Explicit channel information feedback based on high-order PCA decomposition or PCA composition

ABSTRACT

A communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system includes a transceiver to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration, and a processor. The processor estimates the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructs a frequency-domain channel tensor using the CSI estimate, performs a higher-order principal component analysis, HO-PCA, on the channel tensor, identifies a plurality of dominant principal components of the channel tensor, thereby obtaining a compressed channel tensor, and reports to the transmitter the explicit CSI including the dominant principal components of the channel tensor.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of copending International Application No. PCT/EP2019/063887, filed May 28, 2019, which is incorporated herein by reference in its entirety, and additionally claims priority from European Application No. EP 18 175 629.7, filed Jun. 1, 2018, which is incorporated herein by reference in its entirety.

BACKGROUND OF THE INVENTION

The present application concern with the field of wireless communications, more specifically to wireless communication systems employing precoding using explicit channel state information, CSI.

FIG. 1 is a schematic representation of an example of a terrestrial wireless network 100 including a core network 102 and a radio access network 104. The radio access network 104 may include a plurality of base stations gNB₁ to gNB₅, each serving a specific area surrounding the base station schematically represented by respective cells 106 ₁ to 106 ₅. The base stations are provided to serve users within a cell. The term base station, BS, refers to as gNB in 5G networks, eNB in UMTS/LTE/LTE-A/LTE-A Pro, or just BS in other mobile communication standards. A user may be a stationary device or a mobile device. Further, the wireless communication system may be accessed by mobile or stationary IoT devices which connect to a base station or to a user. The mobile devices or the IoT devices may include physical devices, ground based vehicles, such as robots or cars, aerial vehicles, such as manned or unmanned aerial vehicles (UAVs), the latter also referred to as drones, buildings and other items or devices having embedded therein electronics, software, sensors, actuators, or the like as well as network connectivity that enable these devices to collect and exchange data across an existing network infrastructure. FIG. 1 shows an exemplary view of only five cells, however, the wireless communication system may include more such cells. FIG. 1 shows two users UE₁ and UE₂, also referred to as user equipment, UE, that are in cell 106 ₂ and that are served by base station gNB₂. Another user UE₃ is shown in cell 106 ₄ which is served by base station gNB₄. The arrows 108 ₁, 108 ₂ and 108 ₃ schematically represent uplink/downlink connections for transmitting data from a user UE₁, UE₂ and UE₃ to the base stations gNB₂, gNB₄ or for transmitting data from the base stations gNB₂, gNB₄ to the users UE₁, UE₂, UE₃. Further, FIG. 1 shows two IoT devices 110 ₁ and 110 ₂ in cell 106 ₄, which may be stationary or mobile devices. The IoT device 110 ₁ accesses the wireless communication system via the base station gNB₄ to receive and transmit data as schematically represented by arrow 112 ₁. The IoT device 110 ₂ accesses the wireless communication system via the user UE₃ as is schematically represented by arrow 112 ₂. The respective base station gNB₁ to gNB₅ may be connected to the core network 102, e.g. via the S1 interface, via respective backhaul links 114 ₁ to 114 ₅, which are schematically represented in FIG. 1 by the arrows pointing to “core”. The core network 102 may be connected to one or more external networks. Further, some or all of the respective base station gNB₁ to gNB₅ may connected, e.g. via the S1 or X2 interface or XN interface in NR, with each other via respective backhaul links 116 ₁ to 116 ₅, which are schematically represented in FIG. 1 by the arrows pointing to “gNBs”. The wireless network or communication system depicted in FIG. 1 may by an heterogeneous network having two distinct overlaid networks, a network of macro cells with each macro cell including a macro base station, like base station gNB₁ to gNB₅, and a network of small cell base stations (not shown in FIG. 1 ), like femto or pico base stations.

For data transmission a physical resource grid may be used. The physical resource grid may comprise a set of resource elements to which various physical channels and physical signals are mapped. For example, the physical channels may include the physical downlink and uplink shared channels (PDSCH, PUSCH) carrying user specific data, also referred to as downlink and uplink payload data, the physical broadcast channel (PBCH) carrying for example a master information block (MIB) and a system information block (SIB), the physical downlink and uplink control channels (PDCCH, PUCCH) carrying for example the downlink control information (DCI), etc. For the uplink, the physical channels may further include the physical random access channel (PRACH or RACH) used by UEs for accessing the network once a UE synchronized and obtained the MIB and SIB. The physical signals may comprise reference signals (RS), synchronization signals and the like. The resource grid may comprise a frame or radio frame having a certain duration, like 10 milliseconds, in the time domain and having a given bandwidth in the frequency domain. The frame may have a certain number of subframes of a predefined length, e.g., 2 subframes with a length of 1 millisecond. Each subframe may include two slots of 6 or 7 OFDM symbols depending on the cyclic prefix (CP) length. A frame may also consist of a smaller number of OFDM symbols, e.g. when utilizing shortened transmission time intervals (sTTI) or a mini-slot/non-slot-based frame structure comprising just a few OFDM symbols.

Moreover, the downlink signal(s) of the base station (gNB) may contain one or multiple types of RSs including a common RS (CRS) in LTE, a channel state information RS (CSI-RS), a demodulation RS (DM-RS), and a phase tracking RS (PT-RS). The CRS is transmitted over a DL system bandwidth part, and used at the user equipment (UE) to obtain a channel estimate to demodulate the data or control information. The CSI-RS is transmitted with a reduced density in the time and frequency domain compared to CRS, and used at the UE for channel estimation/channel state information (CSI) acquisition. The DM-RS is transmitted only in a bandwidth part of the respective PDSCH and used by the UE for data demodulation. For signal precoding at the gNB, several CSI-RS reporting mechanism were introduced such as non-precoded CSI-RS and beamformed CSI-RS reporting (see reference [1]). For a non-precoded CSI-RS, a one-to-one mapping between a CSI-RS port and a transceiver unit, TXRU, of the antenna array at the gNB is utilized. Therefore, non-precoded CSI-RS provides a cell-wide coverage where the different CSI-RS ports have the same beam-direction and beam-width. For beamformed/precoded UE-specific or non-UE-specific CSI-RS, a beam-forming operation is applied over a single- or multiple antenna ports to have several narrow beams with high gain in different directions and therefore, no cell-wide coverage.

The wireless communication system may be any single-tone or multicarrier system using frequency-division multiplexing, like the orthogonal frequency-division multiplexing (OFDM) system, the orthogonal frequency-division multiple access (OFDMA) system, or any other IFFT-based signal with or without CP, e.g. DFT-s-OFDM. Other waveforms, like non-orthogonal waveforms for multiple access, e.g. filter-bank multicarrier (FBMC), generalized frequency division multiplexing (GFDM) or universal filtered multi carrier (UFMC), may be used. The wireless communication system may operate, e.g., in accordance with the LTE-Advanced pro standard or the 5G or NR, New Radio, standard.

In the wireless communication network as shown in FIG. 1 the radio access network 104 may be a heterogeneous network including a network of primary cells, each including a primary base station, also referred to as a macro base station. Further, a plurality of secondary base stations, also referred to as small cell base stations, may be provided for each of the macro cells. In addition to the above described terrestrial wireless network also non-terrestrial wireless communication networks exist including spaceborne transceivers, like satellites, and/or airborne transceivers, like unmanned aircraft systems. The non-terrestrial wireless communication network or system may operate in a similar way as the terrestrial system described above with reference to FIG. 1 , for example in accordance with the LTE-advanced pro standard or the 5G or NR, new radio, standard.

In a wireless communication system like to one depicted schematically in FIG. 1 , multi-antenna techniques may be used, e.g., in accordance with LTE or NR, to improve user data rates, link reliability, cell coverage and network capacity. To support multi-stream or multi-layer transmissions, linear precoding is used in the physical layer of the communication system. Linear precoding is performed by a precoder matrix which maps layers of data to antenna ports. The precoding may be seen as a generalization of beamforming, which is a technique to spatially direct/focus data transmission towards an intended receiver. The precoder matrix to be used at the gNB to map the data to the transmit antenna ports is decided using channel state information, CSI.

In a wireless communication system employing time division duplexing, TDD, due to channel reciprocity, the channel state information (CSI) is available at the base station (gNB). However, when employing frequency division duplexing, FDD, due to the absence of channel reciprocity, the channel has to be estimated at the UE and feed back to the gNB. FIG. 2 shows a block-based model of a MIMO DL transmission using codebook-based-precoding in accordance with LTE release 8. FIG. 2 shows schematically the base station 200, gNB, the user equipment, UE, 202 and the channel 204, like a radio channel for a wireless data communication between the base station 200 and the user equipment 202. The base station includes an antenna array ANT_(T) having a plurality of antennas or antenna elements, and a precoder 206 receiving a data vector 208 and a precoder matrix F from a codebook 210. The channel 204 may be described by the channel tensor/matrix 212. The user equipment 202 receives the data vector 214 via an antenna or an antenna array ANT_(R) having a plurality of antennas or antenna elements. A feedback channel 216 between the user equipment 202 and the base station 200 is provided for transmitting feedback information. The previous releases of 3GPP up to Rel.15 support the use of several downlink reference symbols (such as CSI-RS) for CSI estimation at the UE. In FDD systems (up to Rel. 15), the estimated channel at the UE is reported to the gNB implicitly where the CSI transmitted by the UE over the feedback channel includes the rank index (RI), the precoding matrix index (PMI) and the channel quality index (CQ) (and the CRI from Rel. 13) allowing, at the gNB, deciding the precoding matrix, and the modulation order and coding scheme (MCS) of the symbols to be transmitted. The PMI and the RI are used to determine the precoding matrix from a predefined set of matrices Ω called ‘codebook’. The codebook, e.g., in accordance with LTE, may be a look-up table with matrices in each entry of the table, and the PMI and RI from the UE decide from which row and column of the table the precoder matrix to be used is obtained. The precoders and codebooks are designed up to Rel. 15 for gNBs equipped with one-dimensional Uniform Linear Arrays (ULAs) having N₁ dual-polarized antennas (in total N_(t)=2N₁ antenna elements), or with two-dimensional Uniform Planar Arrays (UPAs) having N₁N₂ dual-polarized antennas (in total N_(t)=2N₁N₂ antenna elements). The ULA allows controlling the radio wave in the horizontal (azimuth) direction only, so that azimuth-only beamforming at the gNB is possible, whereas the UPA supports transmit beamforming on both vertical (elevation) and horizontal (azimuth) directions, which is also referred to as full-dimension (FD) MIMO. The codebook, e.g., in the case of massive antenna arrays such as FD-MIMO, may be a set of beamforming weights that forms spatially separated electromagnetic transmit/receive beams using the array response vectors of the array. The beamforming weights (also referred to as the ‘array steering vectors’) of the array are amplitude gains and phase adjustments that are applied to the signal fed to the antennas (or the signal received from the antennas) to transmit (or obtain) a radiation towards (or from) a particular direction. The components of the precoder matrix are obtained from the codebook, and the PMI and the RI are used to ‘read’ the codebook and obtain the precoder. The array steering vectors may be described by the columns of a 2D Discrete Fourier Transform (DFT) matrix.

An inherent drawback of implicit feedback is the limited accuracy of the CSI available at the gNB which may be inadequate for the use of advanced precoder techniques such as non-linear (NL) precoding in multi-user settings. Since future NR systems (e.g., Rel. 16) are likely to be based on advanced precoder techniques, the usage of implicit CSI feedback may result in a CSI mismatch which will become a serious issue when high performance gains are targeted. Considering this issue, RAN1 has agreed to support a specification on advanced CSI reporting such as explicit CSI in the upcoming Rel. 16. Here, explicit CSI refers to reporting of explicit channel coefficients from the UE to the gNB without a codebook for the precoder selection at the UE. With explicit CSI feedback, no codebook is used to determine the precoder. The coefficients of the precoder matrix are transmitted explicitly by the UE. Alternatively, the coefficients of the instantaneous channel matrix may be transmitted, from which the precoder is determined by the gNB.

WO 2018/052255 A1 relates to explicit CSI acquisition to represent the channel in wireless communication systems using the principal component analysis (PCA), which is applied on the frequency-domain channel matrix, covariance matrix, or eigenvector of the channel matrix. A major disadvantage of the PCA approach is that the size (with respect to the dimensions) of the “compressed” channel matrix after the PCA decomposition is identical to the size of the “uncompressed” channel matrix. Therefore, the feedback overhead scales linearly with the system bandwidth size, i.e., with the total number of allocated sub-bands. Thus, the availability of accurate explicit CSI comes at an increased overhead for the feedback channel which is not desired.

This issue could be solved, in principle, with an increase in the sub-band size for large system bandwidth sizes. However, the increase of the sub-band size comes at the expense of a reduced CSI accuracy at the gNB.

It is noted that the information in the above section is only for enhancing the understanding of the background of the invention and therefore it may contain information does not form known technology that is already known to a person of ordinary skill in the art.

SUMMARY

An embodiment may have a communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device including: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration, and a processor configured to

-   -   estimate the CSI using measurements on the downlink reference         signals of the radio channel according to the reference signal         configuration over one or more time instants/slots,     -   construct a frequency-domain channel tensor using the CSI         estimate,     -   perform a higher-order principal component analysis, HO-PCA, on         the channel tensor,     -   identify a plurality of dominant principal components of the         channel tensor, thereby acquiring a compressed channel tensor,         and     -   report to the transmitter the explicit CSI including the         dominant principal components of the channel tensor.

Another embodiment may have a communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device including: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration; and a processor configured to

-   -   estimate the CSI using measurements on the downlink reference         signals of the radio channel according to the reference signal         configuration over one or more time instants/slots,     -   construct a frequency-domain channel tensor using the CSI         estimate,     -   calculate a transformed channel tensor using the channel tensor,     -   rewrite the transformed channel tensor to a transformed channel         matrix,     -   perform a standard principal component analysis, PCA, on the         transformed channel matrix,     -   identify a plurality of dominant principal components of the         transformed channel matrix,     -   thereby acquiring a transformed/compressed channel matrix, and     -   report to the transmitter the explicit CSI including the         plurality of dominant principal components of the         transformed/compressed channel tensor.

Another embodiment may have a communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device including: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration; and a processor configured to

-   -   estimate the CSI using measurements on the downlink reference         signals of the radio channel,     -   construct a frequency-domain channel matrix using the CSI         estimate,     -   perform a standard principal component analysis, PCA, on the         channel matrix,     -   identify the dominant principal components of the channel         matrix, the channel matrix including         -   a first set of r basis vectors contained in a matrix Ū=[u₁,             u₂, . . . , u_(r)]∈             ^(N) ^(t) ^(N) ^(r) ^(×r).         -   a second set of r basis vectors contained in a matrix V=[v₁,             v₂, . . . , v_(r)]∈             ^(S×r); and         -   a third set of r coefficients contained in a diagonal matrix             Σ=diag(s₁, s₂, . . . , s_(r)) ∈             ^(r×r) with ordered singular values s_(i) (s₁≥s₂≥ . . .             ≥s_(r)) on its diagonal;     -   calculate a reduced-sized delay-domain matrix {tilde over (V)}         from the frequency-domain matrix V, wherein the delay-domain         matrix, including r basis vectors, is given by         V≈F _(V) {tilde over (V)},         -   where             -   F_(V)∈                 ^(S×L) contains L vectors of size S×1, selected from a                 discrete Fourier transform, DFT, codebook Ω, the size of                 the compressed delay-domain matrix {tilde over                 (V)}=[{tilde over (v)}₁, {tilde over (v)}₂, . . . ,                 {tilde over (v)}_(r)] is given by L×r, and L is the                 number of delays, and         -   the oversampled codebook matrix is given by Ω=[d₀, d₁, . . .             , d_(SO) _(f) −1], where

${\left\lbrack {1\mspace{14mu} e^{\frac{{- j}\; 2\pi\; i}{O_{f}S}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\pi\;{i{({S - 1})}}}{O_{f}S}}} \right\rbrack^{T} \in {\mathbb{C}}^{S \times 1}},$ i∈{0, . . . , SO_(f)−1}, and 0_(f)∈{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; and

-   -   report to the transmitter the explicit CSI including the         identified first set of r basis vectors, the second         reduced-sized delay-domain set of r basis vectors, along with         the L delays, represented by a set of indices that correspond to         the selected DFT-vectors in the codebook Ω.

Another embodiment may have a transmitter in a wireless communication system, the transmitter including: an antenna array including a plurality of antennas for a wireless communication with one or more communication devices according to the invention for providing a channel state information, CSI, feedback to the transmitter; and a precoder connected to the antenna array, the precoder to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams, a transceiver configured to transmit, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS, and downlink signals including the CSI-RS configuration; and receive uplink signals including a plurality of CSI reports including an explicit CSI from the communication device; and a processor configured to construct a precoder matrix applied on the antenna ports using the explicit CSI.

Another embodiment may have a wireless communication network, including: at least one communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device including: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration, and a processor configured to

-   -   estimate the CSI using measurements on the downlink reference         signals of the radio channel according to the reference signal         configuration over one or more time instants/slots,     -   construct a frequency-domain channel tensor using the CSI         estimate,     -   perform a higher-order principal component analysis, HO-PCA, on         the channel tensor,     -   identify a plurality of dominant principal components of the         channel tensor, thereby acquiring a compressed channel tensor,         and         report to the transmitter the explicit CSI including the         dominant principal components of the channel tensor, and         at least one transmitter according to the invention.

Another embodiment may have a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method having the steps of: receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, performing a high-order principal component analysis, HO-PCA, on the channel tensor, identifying a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and reporting the explicit CSI including the dominant principal components of the channel tensor from the communication device to the transmitter.

Another embodiment may have a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method having the steps of: receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, calculating a transformed channel tensor using the channel tensor, rewriting the transformed channel tensor to a transformed channel matrix, performing a standard principal component analysis, PCA, on the transformed channel matrix, identifying a plurality of dominant principal components of the transformed channel matrix, thereby acquiring a transformed/compressed channel matrix, and reporting the explicit CSI including the plurality of dominant principal components of the transformed/compressed channel tensor from the communication device to the transmitter.

Another embodiment may have a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method having the steps of: receiving, from a transmitter, a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration; and estimating the CSI using measurements on the downlink reference signals of the radio channel, constructing a frequency-domain channel matrix using the CSI estimate, performing a standard principal component analysis, PCA, on the channel matrix, identifying the dominant principal components of the channel matrix, the channel matrix including

-   -   a first set of r basis vectors contained in a matrix Ū=[u₁, u₂,         . . . , u_(r)]∈         ^(N) ^(t) ^(N) ^(r) ^(×r);     -   a second set of r basis vectors contained in a matrix V=[v₁, v₂,         . . . , v_(r)]∈         ^(S×r); and     -   a third set of r coefficients contained in a diagonal matrix         Σ=diag(s₁, s₂, . . . , s_(r))∈         ^(r×r) with ordered singular values s_(i) (s₁≥s₂ . . . ≥s_(R))         on its diagonal;     -   calculating a reduced-sized delay-domain matrix {tilde over (V)}         from the frequency-domain matrix V, wherein the delay-domain         matrix, including r basis vectors, is given by         V≈F _(V) {tilde over (V)},     -   where         -   F_(V)∈             ^(S×L) contains L vectors of size S×1, selected from a             discrete Fourier transform, DFT, codebook Ω, the size of the             compressed delay-domain matrix {tilde over (V)}=[{tilde over             (v)}₁, {tilde over (v)}₂, . . . , {tilde over (v)}_(r)] is             given by L×r, and L is the number of delays, and         -   the oversampled codebook matrix is given by Ω=[d₀, d₁, . . .             , d_(SO) _(f) ⁻¹], where

${\left\lbrack {1\mspace{14mu} e^{\frac{{- j}\; 2\pi\; i}{O_{f}S}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\pi\;{i{({S - 1})}}}{O_{f}S}}} \right\rbrack^{T} \in {\mathbb{C}}^{S \times 1}},$ i∈{0, . . . , SO_(f)−1}, and O_(f)∈{1, 2, 3, . . . }denotes the oversampling factor of the DFT-codebook matrix; and

-   -   reporting the explicit CSI including the identified first set of         r basis vectors, the second reduced-sized delay-domain set of r         basis vectors, along with the L delays, represented by a set of         indices that correspond to the selected DFT-vectors in the         codebook Ω from the communication device to the transmitter.

Another embodiment may have a method for transmitting in a wireless communication system including a communication device a communication device of claim 1 and a transmitter, the method having the steps of: transmitting, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS configuration, and downlink signals including the CSI-RS configuration; receiving, at the transmitter, uplink signals including a plurality of CSI reports including an explicit CSI from the communication device; constructing a precoder matrix for a precoder connected to an antenna array including a plurality of antennas; applying the precoder matrix on antenna ports using the explicit CSI so as to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams.

Another embodiment may have a non-transitory digital storage medium having a computer program stored thereon to perform the method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method having the steps of: receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, performing a high-order principal component analysis, HO-PCA, on the channel tensor, identifying a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and reporting the explicit CSI including the dominant principal components of the channel tensor from the communication device to the transmitter; when said computer program is run by a computer.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will be detailed subsequently referring to the appended drawings, in which:

FIG. 1 shows a schematic representation of an example of a wireless communication system;

FIG. 2 shows a block-based model of a MIMO DL transmission using codebook-based-precoding in accordance with LTE release 8;

FIG. 3 is a schematic representation of a wireless communication system for communicating information between a transmitter, which may operate in accordance with the inventive teachings described herein, and a plurality of receivers, which may operate in accordance with the inventive teachings described herein;

FIG. 4 is a flow diagram illustrating a HO-PCA decomposition/compression of a channel tensor, the reporting at a UE and the reconstruction of the channel tensor at the gNB in accordance with an embodiment of the present invention;

FIG. 5 illustrates a frequency-domain channel tensor (three-dimensional array)

of dimension N_(r)×N_(t)×S;

FIG. 6 is a flow diagram illustrating a HO-PCA decomposition/compression of a channel tensor in combination with delay-domain compression, the reporting at the UE and the reconstruction of the channel tensor at gNB in accordance with an embodiment of the present invention;

FIG. 7 is a flow diagram illustrating a PCA decomposition/compression of a channel matrix, the reporting at a UE and the reconstruction of the channel matrix at the gNB in accordance with an embodiment of the present invention;

FIG. 8 illustrates a frequency-domain channel matrix (two-dimensional array) H of dimension N×S, where N=N_(t)N_(r);

FIG. 9 is a flow diagram illustrating a transformation/compression of a channel tensor in combination with a HO-PCA decomposition, the reporting at the UE and the reconstruction of the channel tensor at gNB in accordance with an embodiment of the present invention;

FIG. 10 is a flow diagram illustrating a transformation/compression of a channel matrix in combination with a non-HO PCA decomposition, the reporting at the UE and the reconstruction of channel tensor at the gNB in accordance with an embodiment of the present invention;

FIG. 11 is a flow diagram illustrating a HO-PCA decomposition/compression of a four-dimensional channel tensor, the reporting at the UE and the reconstruction of channel tensor at the gNB in accordance with an embodiment of the present invention;

FIG. 12 illustrates a frequency domain channel matrix of size N×SD, where N=N_(t)N_(r);

FIG. 13 is a flow diagram illustrating a HO-PCA decomposition/compression of a four-dimensional channel tensor in combination with a delay- or time/doppler domain compression, the reporting at the UE and the reconstruction the of channel tensor at the gNB in accordance with an embodiment of the present invention;

FIG. 14 is a flow diagram illustrating a transformation/compression of a four-dimensional channel tensor in addition to a HO-PCA decomposition, the reporting at the UE and the reconstruction of the channel tensor at the gNB in accordance with an embodiment of the present invention;

FIG. 15 is a flow diagram illustrating a transformation/compression of a four-dimensional channel tensor in addition to non-HO-PCA decomposition, the reporting at the UE and the reconstruction of the channel matrix at the gNB in accordance with an embodiment of the present invention; and

FIG. 16(a) illustrates a CSI-RS with a periodicity of 10 slots and no repetition (CSI-RS-BurstDuration not configured or CSI-RS-BurstDuration=0);

FIG. 16(b) illustrates a CSI-RS with a periodicity of 10 slots and repetition of 4 slots (CSI-RS-BurstDuration=4);

FIG. 17 illustrates a CSI-RS-BurstDuration information element in accordance with an embodiment;

FIG. 18 illustrates an example of a computer system on which units or modules as well as the steps of the methods described in accordance with the inventive approach may execute.

DETAILED DESCRIPTION OF THE INVENTION

In the following, embodiments of the present invention are described in further detail with reference to the enclosed drawings in which elements having the same or similar function are referenced by the same reference signs.

Embodiments of the present invention may be implemented in a wireless communication system or network as depicted in FIG. 1 or FIG. 2 including transmitters or transceivers, like base stations, and receivers or users, like mobile or stationary terminals or IoT devices, as mentioned above. FIG. 3 is a schematic representation of a wireless communication system for communicating information between a transmitter 200, like a base station, and a plurality of receivers 202 ₁ to 202 _(n), like UEs, which are served by the base station 200. The base station 200 and the UEs 202 may communicate via a wireless communication link or channel 204, like a radio link. The base station 200 includes one or more antennas ANT_(T) or an antenna array having a plurality of antenna elements, and a signal processor 200 a. The UEs 202 include one or more antennas ANT_(R) or an antenna array having a plurality of antennas, a signal processor 202 a ₁, 202 a _(n), and a transceiver 202 b ₁, 202 b _(n). The base station 200 and the respective UEs 202 may operate in accordance with the inventive teachings described herein.

Communication Device

The present invention provides a communication device 202 for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising a transceiver 202 b configured to receive, from a transmitter 200 a radio signal via a radio time-variant frequency MIMO channel 204, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, and a processor 202 a configured to

-   -   estimate the CSI using measurements on the downlink reference         signals of the radio channel according to the reference signal         configuration over one or more time instants/slots,     -   construct a frequency-domain channel tensor using the CSI         estimate,     -   perform a higher-order principal component analysis, HO-PCA, on         the channel tensor,     -   identify a plurality of dominant principal components of the         channel tensor, thereby obtaining a compressed channel tensor,         and     -   report to the transmitter 200 the explicit CSI comprising the         dominant principal components of the channel tensor.

In accordance with embodiments, the communication device is configured to receive from the transmitter an explicit CSI report configuration containing a CSI channel-type information, CSI-Ind, indicator for the CSI report, wherein the CSI-Ind indicator is associated with a channel-type configuration, the channel tensor is a three-dimensional, 3D, channel tensor, or is represented by a 3D channel covariance tensor, a 3D beamformed-channel tensor, or a 3D beamformed channel covariance tensor, as indicated by the CSI-Ind indicator, and wherein the plurality of dominant principal components of the compressed 3D channel tensor of dimension N_(r)×N_(t)×S comprise:

-   -   a first set of r₁ basis vectors contained in a matrix         Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ]∈         ^(N) ^(r) ^(×r) ¹ ;     -   a second set of r₂ basis vectors contained in a matrix         Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ]∈         ^(N) ^(t) ^(×r) ² ;     -   a third set of r₃ basis vectors contained in a matrix         Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ]∈         ^(S×r) ³ ; and     -   r₁r₂r₃associated high-order singular values s_(ijk), sorted such         that Σ_(j,k)|s_(i,j,k)|²≥Σ_(j,k)|s_(i+1,j,k)|²,         Σ_(i,k)|s_(i,j,k)|²≥Σ_(i,k)|s_(i,j+1,k)|², and         Σ_(i,j)|²≥Σ_(i,j)|s_(i,j,k+1)|² for all i,j,k.

In accordance with embodiments, the values of r₁, r₂, and r₃ representing the number of dominant principal components with respect to the first, second and third dimension of the compressed 3D channel tensor, respectively, are

-   -   configured via the CSI report configuration by the transmitter,         or     -   reported by the communication device in the CSI report, or     -   pre-determined and known at the communication device.

In accordance with embodiments, the processor is configured to quantize the coefficients in the vectors u_(R,i), u_(T,j), and u_(S,k) and the HO singular values s_(ijk) of the 3D channel tensor using a codebook approach, the number of complex coefficients to be quantized being given by N_(r)r₁+N_(t)r₂+Sr₃ for the higher-order singular vectors, and a number of real coefficients to be quantized being given by r₁r₂r₃ for the higher-order singular values, respectively.

In accordance with embodiments, the channel tensor is a four-dimensional, 4D, channel tensor, or represented either by a 4D channel covariance tensor, a 4D beamformed-channel tensor, or a 4D beamformed channel covariance tensor, as indicated by the CSI-Ind indicator, and the plurality of dominant principal components of the compressed 4D channel tensor of dimension N_(r)×N_(t)×S×D comprise:

-   -   a first set of r₁ basis vectors contained in a matrix Ū_(R)         [u_(R,1), . . . , u_(R,r) ₁ ]∈         ^(N) ^(r) ^(×r) ¹ ;     -   a second set of r₂ basis vectors contained in a matrix         Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ]∈         ^(N) ^(t) ^(×r) ² ;     -   a third set of r₃ basis vectors contained in a matrix         Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ]∈         ^(S×r) ³ ;     -   a fourth set of r₄ basis vectors contained in a matrix         Ū_(D)=[u_(D,1), . . . , u_(D,r) ₄ ]∈         ^(D×r) ⁴ ; and     -   r₁r₂r₃r₄ associated high-order singular values s_(ijkl), sorted         such that         Σ_(j,k,l)|s_(i,j,k,l)|²≥Σ_(j,k,l)|s_(i+1,j,k,l)|²,Σ|s_(i,j,k,l)|²≥Σ_(i,k,l)|²,Σ_(i,j,l)|s_(i,j,k,l)|²≥Σ_(i,j,l)|s_(i,j,k+1,l)|²,         and Σ_(i,j,k)|²≥Σ_(i,j,k)|s_(i,j,k,l+1)|² for all i, j, k, l.

In accordance with embodiments, the values of r₁, r₂, r₃, and r₄ representing the number of dominant principal components of the 4D channel tensor are configured via the CSI report configuration by the transmitter, or they are reported by the communication device in the CSI report, or they are pre-determined and known at the communication device.

In accordance with embodiments, the processor is configured to quantize the coefficients of the vectors u_(R,i), u_(T,j), u_(S,k), u_(D,l) and the singular values s_(ijkl) using a codebook approach, a number of complex coefficients to be quantized being given by N_(r)r₁+N_(t)r₂+Sr₃+Dr₄ for the higher-order singular vectors, and a number of real coefficients to be quantized being given by r₁r₂r₃r₄ for the higher-order singular values, respectively.

In accordance with embodiments, the explicit CSI comprises a delay-domain CSI for the higher-order singular-value matrix Ū_(S), and wherein the processor is configured to calculate a reduced-sized delay-domain higher-order singular-matrix Ũ_(S) from the frequency-domain higher-order singular-matrix Ū_(S), wherein the delay-domain higher-order singular-matrix is given by Ū _(S) ≈F _(S) ŨS, where

-   -   F_(S)∈         ^(S×L) contains L vectors of size S×1, selected from a discrete         Fourier transform, DFT, codebook Ω, the size of the compressed         delay-domain matrix Ũ_(S) is given by L×r₃, and L is the number         of delays, and     -   the oversampled codebook matrix is given by Ω=[d₀, d₁, . . . ,         d_(SO) _(f) ⁻¹], where d_(i)=

${\left\lbrack {1\mspace{14mu} e^{\frac{{- j}\; 2\pi\; i}{O_{f}S}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\pi\;{i{({S - 1})}}}{O_{f}S}}} \right\rbrack^{T} \in {\mathbb{C}}^{S \times 1}},$ i∈{0, . . . , SO_(f)−1}, and O_(f)∈{1,2,3, . . . } denotes the oversampling factor of the DFT-codebook matrix; quantize the coefficients in vectors Ũ_(S)=[ũ_(S,1), . . . , ũ_(S,r) ₃ ]∈

^(L×r) ³ using a codebook approach; report to the transmitter the explicit CSI containing the coefficients of Ũ_(S) instead of Ū_(S), along with the L delays, represented by a set of indices that correspond to the selected DFT-vectors in the codebook Ω.

In accordance with embodiments, the explicit CSI comprises a Doppler-frequency domain CSI for the higher-order singular-value matrices Ū_(D), and wherein the processor is configured to

calculate a reduced-sized Doppler-frequency domain higher-order singular-matrix Ũ_(D) from the time-domain higher-order singular-matrix Ū_(D), wherein the Doppler-frequency domain higher-order singular-matrix is given by Ū _(D) ≈F _(D) Ũ _(D), where

-   -   F_(D)∈         ^(D×G) contains G vectors of size D×1, selected from a discrete         Fourier transform, DFT, codebook Ω, the size of the compressed         Doppler-frequency domain matrix Ũ_(D) is given by G×r₄, and G is         the number of Doppler-frequency components, and     -   the oversampled codebook matrix is given by Ω=[d₀, d₁, . . . ,         d_(DO) _(t) ⁻¹], where d_(i)=

${\left\lbrack {1\mspace{14mu} e^{\frac{{- j}\; 2\pi\; i}{O_{f}D}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\pi\;{i{({D - 1})}}}{O_{f}D}}} \right\rbrack^{T} \in {\mathbb{C}}^{D \times 1}},$ i∈{0, . . . , O_(t)D−1}, and O_(t)∈{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; quantize the coefficients in vectors Ũ_(D)=[ũ_(D,1), . . . , ũ_(D,r) ₄ ]∈

^(G×r) ⁴ using a codebook approach; or report to the transmitter the explicit CSI report containing the coefficients of Ũ_(D) instead of Ū_(D), along with the G Doppler-frequency components, represented by a set of indices that correspond to the selected DFT-vectors in the codebook Ω.

The present invention provides a communication device 202 for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising a transceiver 202 b configured to receive, from a transmitter 200 a radio signal via a radio time-variant frequency MIMO channel 204, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and a processor 202 a configured to

-   -   estimate the CSI using measurements on the downlink reference         signals of the radio channel according to the reference signal         configuration over one or more time instants/slots,     -   construct a frequency-domain channel tensor using the CSI         estimate,     -   calculate a transformed channel tensor using the channel tensor,     -   rewrite the transformed channel tensor to a transformed channel         matrix,     -   perform a standard principal component analysis, PCA, on the         transformed channel matrix,     -   identify a plurality of dominant principal components of the         transformed channel matrix,     -   thereby obtaining a transformed/compressed channel matrix, and     -   report to the transmitter the explicit CSI comprising the         plurality of dominant principal components of the         transformed/compressed channel tensor.

In accordance with embodiments, the communication device is configured to receive from the transmitter an explicit CSI report configuration containing a CSI channel-type information, CSI-Ind, indicator for the CSI report, wherein the CSI-Ind indicator is associated with a channel-type configuration, the channel tensor is a three-dimensional, 3D, channel tensor, or is represented either by a 3D channel covariance tensor, a 3D beamformed-channel tensor, or a 3D beamformed channel covariance tensor as indicated by the CSI-Ind indicator, or the channel tensor is a four-dimensional, 4D, channel tensor, or is represented either by a 4D channel covariance tensor, a 4D beamformed-channel tensor, or a 4D beamformed channel covariance tensor as indicated by the CSI-Ind indicator, and the 3D transformed channel tensor of dimension N′_(r)×N′_(t)×S′, or the 4D transformed channel tensor of dimension N′_(t)×N′_(t)×S′×D′, is rewritten to a transformed channel matrix of dimension N′_(t)N′_(r)×S′D′, where D′=1 for the 3D transformed channel tensor and D′>1 for the 4D transformed channel tensor, respectively, and the plurality of dominant principal components of the transformed channel matrix comprise:

-   -   a first set of r basis vectors contained in a matrix Ū=[u₁, u₂,         . . . , u_(r)]∈         ^(N′) ^(t) ^(N′) ^(r) ^(×r);     -   a second set of r basis vectors contained in a matrix V=[v₁, v₂,         . . . , v_(r)]∈         ^(S′D′×r).     -   a set of r coefficients contained in a diagonal matrix         Σ=diag(s₁, s₂, . . . , s_(r)) ∈         ^(r×r) with ordered singular values s_(i) (s₁≥s₂≥, . . . ,         s_(r)) on its diagonal.

In accordance with embodiments, the value of r representing the number of dominant principal components of the transformed channel matrix is

-   -   configured via the CSI report configuration by the transmitter,         or     -   reported by the communication device in the CSI report, or     -   pre-determined and known at the communication device.

In accordance with embodiments, the processor is configured to quantize the coefficients of the basis vectors u_(i), v_(i), and the singular values s_(i) of the transformed channel matrix using a codebook approach.

In accordance with embodiments, the processor is configured to apply,

after the construction of the 3D channel tensor, a one-dimensional, a two-dimensional, or multi-dimensional transformation/compression of the channel tensor with respect to the space dimension of the 3D channel tensor, or the frequency dimension of the 3D channel tensor, or both the frequency and space dimensions of the 3D channel tensor, or after the construction of the 4D channel tensor, a one-dimensional, a two-dimensional, or multi-dimensional transformation/compression of the channel tensor with respect to the space dimension of the channel tensor, or the frequency dimension of the channel tensor, or the time dimension of the channel tensor, or both the frequency and time dimensions of the channel tensor, so as to exploit a sparse or nearly-sparse representation of the 3D or 4D channel tensor in one or more dimensions.

In accordance with embodiments, the processor is configured to apply, after the construction of the 3D channel tensor, a transformation/compression with respect to

all three dimensions of the 3D channel tensor

of dimension N_(r)×N_(t)×S represented by a (column-wise) Kronecker product as

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{({1,2,3})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2,3})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,n_{r},n_{t},s} \otimes b_{2,n_{r},n_{t},s} \otimes b_{1,n_{r},n_{t},s}}}}}}}},}$ where

-   -   b_(1,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         N_(r)×1 with respect to the first dimension of the 3D channel         tensor         , selected from a codebook matrix Ω₁;     -   b_(2,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         N_(t)×1 with respect to the second dimension of the 3D channel         tensor         , selected from a codebook matrix Ω₂;     -   b_(3,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         S×1 with respect to the third dimension of the 3D channel tensor         , selected from a codebook matrix Ω₃;     -   γ_(n) _(r) _(,n) _(t) _(,s) is the transformed/compressed         channel coefficient associated with the vectors b_(1,n) _(r)         _(,n) _(t) _(,s), b_(2,n) _(r) _(,n) _(t) _(,s), and b_(3,n)         _(r) _(,n) _(t) _(,s), and     -   N′_(r), N′_(t), and S′ (N′_(r)≤N_(r), N′_(t)≤N_(t), S′≤S)         represents the value of the first, second and third dimension of         the transformed/compressed 3D channel tensor         , respectively, or         the space dimensions of the 3D channel tensor, represented by

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,s} \otimes b_{2,n_{r},n_{t},s} \otimes b_{1,n_{r},n_{t},s}}}}}}}},}$ where

-   -   b_(3,s) is a transformation vector of all zeros with the s-th         element being one,     -   b_(2,n) _(r) _(,n) _(t) _(,s) is a vector of size N_(t)×1 with         respect to the second dimension of the 3D channel tensor         , selected from a codebook matrix Ω₂, and     -   b_(1,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         N_(r)×1 with respect to the first dimension of the 3D channel         tensor         , selected from a codebook matrix Ω₁, N′_(r)≤N_(r),         N′_(t)≤N_(t), S′=S, or         the frequency dimension of the 3D channel tensor, represented by

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,n_{r},n_{t},s} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}$ where

-   -   b_(2,n) _(r) is a transformation vector of all zeros with the         n_(r)-th element being one, b_(1,n) _(t) is a vector of all         zeros with the n_(t)-th element being one, and     -   b_(3,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         S×1 with respect to the third dimension of the 3D channel tensor         , selected from a codebook matrix Ω₃, N′_(r)=N_(r) and         N′_(t)=N_(t), S′≤S.

In accordance with embodiments, (i) the 3D transformation function

_((1,2,3)))(·) is given by a two-dimensional Discrete Cosine transformation (2D-DCT) and a 1D-DFT transformation with respect to the space and frequency dimension of the channel tensor, respectively, the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices, and the codebook matrix Ω₃ is given by an oversampled DFT matrix, or (ii) the 3D transformation function

_((1,2,3))(·) is given by a 3D-DFT transformation and the codebook matrices Ω_(n), n=1,2,3 are given by oversampled DFT matrices, or (iii) the 2D transformation function

_((1,2))(·) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices, or (iv) the 1D transformation function

₍₃₎(·) is given by a 1D-DFT transformation and the codebook matrix Ω₃ is given by an oversampled DFT matrix.

In accordance with embodiments, the processor is configured to apply, after the construction of the 4D channel tensor a transformation/compression with respect to

all four dimensions of the channel tensor, represented by a (column-wise) Kronecker product as

=

_((1,2,3,4))(

),

${{{vec}(\mathcal{H})} = {{{\nu ec}\left( {\mathcal{F}_{({1,2,3,4})}\left( \overset{\hat{}}{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{{\gamma\;}_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r},n_{t},s,d} \otimes b_{1,n_{r},n_{t},s,d}}}}}}}}},$ where

-   -   b_(1,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size N_(r)×1 with respect to the first dimension of the channel         tensor         , selected from a codebook matrix Ω₁;     -   b_(2,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size N_(t)×1 with respect to the second dimension of the channel         tensor         , selected from a codebook matrix Ω₂;     -   b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size S×1 with respect to the third dimension of the channel         tensor         , selected from a codebook matrix Ω₃;     -   b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size D×1 with respect to the fourth dimension of the channel         tensor         , selected from a codebook matrix Ω₄;     -   γ_(n) _(r) _(,n) _(t) _(,s,d) is the transformed/compressed         channel coefficient associated with the vectors b_(1,n) _(r)         _(,n) _(t) _(,s,d), b_(2,n) _(r) _(,n) _(t) _(,s,d), b_(3,n)         _(r) _(,n) _(t) _(,s,d), and b_(4,n) _(r) _(,n) _(t) _(,s,d) and     -   N′_(r), N′_(t), S′ and D′ (N′_(r)≤N_(r), N′_(t)≤N_(t), S′≤S,         D′≤D) represents the value of the first, second, third and         fourth dimension of the transformed/compressed channel tensor         , respectively, or         the space dimensions of the 4D channel tensor, represented by         =         _((1,2))(         ),

${{{vec}(\mathcal{H})} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{{\gamma\;}_{n_{r},n_{t},s,d}{b_{4,d} \otimes b_{3,s} \otimes b_{2,n_{r},n_{t},s,d} \otimes b_{1,n_{r},n_{t},s,d}}}}}}}},$ where

-   -   b_(4,d) is a transformation vector of all zeros with the d-th         element being one,     -   b_(3,s) is a transformation vector of all zeros with the s-th         element being one,     -   b_(2,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size N_(t)×1 with respect to the second dimension of the channel         tensor         , selected from a codebook matrix Ω₂, and     -   b_(1,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size N_(r)×1 with respect to the first dimension of the channel         tensor         , selected from a codebook matrix Ω₁ and N′_(r)≤N_(r),         N′_(t)≤N_(t), S′=S and D′=D, or         the frequency and time dimensions of the 4D channel tensor,         represented by         =         _((3,4))(         ),

${{{vec}(\mathcal{H})} = {{{\nu ec}\left( {\mathcal{F}_{({3,4})}\left( \overset{\hat{}}{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{{\gamma\;}_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}}},$ where

-   -   b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size D×1 with respect to the fourth dimension of the channel         tensor         ,selected from a codebook matrix Ω₄,     -   b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size S×1 with respect to the third dimension of the channel         tensor         , selected from a codebook matrix Ω₃,     -   b_(2,n) _(r) is a transformation vector of all zeros with the         n_(r)-th element being one, and     -   b_(1,n) _(t) is a transformation vector of all zeros with the         n_(t)-th element being one and N′_(r)=N_(r) N′_(t)=N_(t), S′≤S         and D′≤D.

In accordance with embodiments, (i) the 4D transformation function

_((1,2,3,4))(·) is given by a 4D-DFT transformation and the codebook matrices Ω_(n), n=1,2,3,4 are given by oversampled DFT matrices, or (ii) the 2D transformation/compression function

_((3,4))(·) is given by a 2D-DFT and the codebook matrices Ω_(n), n=3,4 are given by oversampled DFT matrices, or (iii) the 2D transformation function

_((1,2))(·) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices, or (iv) the 1D transformation function

₍₃₎(·) is given by a 1D-DFT transformation and the codebook matrix Ω₃ is given by an oversampled DFT matrix, or (v) the 1D transformation function

₍₄₎(·) is given by a 1D-DFT transformation and the codebook matrix Ω₄ is given by an oversampled DFT matrix.

In accordance with embodiments, the processor is configured to

select the transformation vectors from codebook matrices Q_(n), and store the selected indices in a set

of g-tuples, where g refers to the number of transformed dimensions of the channel tensor, and

report the set

as a part of the CSI report to the transmitter.

In accordance with embodiments, the oversampling factors of the codebooks are

-   -   configured via the CSI report configuration, or via higher layer         or physical layer by the transmitter,     -   pre-determined and known at the communication device.

The present invention provides a communication device 202 for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising a transceiver 202 b configured to receive, from a transmitter 200 a radio signal via a radio time-variant frequency MIMO channel 204, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and a processor 202 a configured to

-   -   estimate the CSI using measurements on the downlink reference         signals of the radio channel,     -   construct a frequency-domain channel matrix using the CSI         estimate,     -   perform a standard principal component analysis, PCA, on the         channel matrix,     -   identify the dominant principal components of the channel         matrix, the channel matrix comprising         -   a first set of r basis vectors contained in a matrix Ū=[u₁,             u₂, . . . , u_(r)]∈             ^(N) ^(t) ^(N) ^(r) ^(×r);         -   a second set of r basis vectors contained in a matrix V=[v₁,             v₂, . . . , v_(r)]∈             ^(S×r); and         -   a third set of r coefficients contained in a diagonal matrix             Σ=diag(s₁, s₂, . . . , s_(r))∈             ^(r×r) with ordered singular values s_(i) (s₁≥s₂≥ . . .             ≥s_(r)) on its diagonal;     -   calculate a reduced-sized delay-domain matrix {tilde over (V)}         from the frequency-domain matrix V, wherein the delay-domain         matrix, comprising r basis vectors, is given by         V≈F _(V) {tilde over (V)},         where     -   F_(V)∈         ^(S×L) contains L vectors of size S×1, selected from a discrete         Fourier transform, DFT, codebook Ω, the size of the compressed         delay-domain matrix {tilde over (V)}=[{tilde over (v)}₁, {tilde         over (v)}₂, . . . , {tilde over (v)}_(r)] is given by L×r, and L         is the number of delays, and     -   the oversampled codebook matrix is given by Ω=[d₀, d₁, . . . ,         d_(SO) _(f) ⁻¹], where

${\left\lbrack {1\mspace{14mu} e^{\frac{{- j}\; 2\pi\; i}{O_{f}S}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\pi\;{i{({S - 1})}}}{O_{f}S}}} \right\rbrack^{T} \in {\mathbb{C}}^{S \times 1}},$ i∈{0, . . . , SO_(f)−1}, and O_(f)∈{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; and

-   -   report to the transmitter the explicit CSI containing the         identified first set of r basis vectors, the second         reduced-sized delay-domain set of r basis vectors, along with         the L delays, represented by a set of indices that correspond to         the selected DFT-vectors in the codebook Ω.

In accordance with embodiments, the value of r representing the number of dominant principal components of the transformed channel matrix is configured via the CSI report configuration by the transmitter, or it is reported by the communication device in the CSI report, or it is pre-determined and known at the communication device.

In accordance with embodiments, the processor is configured to quantize the coefficients of the basis vectors u_(i), {tilde over (v)}_(i), and the singular values s_(i) of the channel matrix using a codebook approach.

In accordance with embodiments, the communication device is configured with

-   -   one or more scalar codebooks for the quantization of each entry         of each basis vector of the plurality of dominant principal         components of the channel tensor or the compressed channel         tensor and the singular values or high order singular values, or     -   with one or more unit-norm vector codebooks for the quantization         of each basis vector of the plurality of dominant principal         components of the channel tensor or the compressed channel         tensor, and wherein the communication device selects for each         basis vector a codebook vector to represent the basis vector,         and         the communication device is configured to report the indices         corresponding to the selected entries in the scalar or vector         codebook as a part of the CSI report to the transmitter.         Transmitter

The present invention provides a transmitter 200 in a wireless communication system, the transmitter comprising:

-   -   an antenna array ANT_(T) having a plurality of antennas for a         wireless communication with one or more of the inventive         communication devices 202 a, 202 b for providing a channel state         information, CSI, feedback to the transmitter 200; and a         precoder 206 connected to the antenna array ANT_(T), the         precoder 206 to apply a set of beamforming weights to one or         more antennas of the antenna array ANT_(T) to form, by the         antenna array ANT_(T), one or more transmit beams or one or more         receive beams,         a transceiver 200 b configured to     -   transmit, to the communication device 202 a, 202 b, downlink         reference signals (CSI-RS) according to a CSI-RS, and downlink         signals comprising the CSI-RS configuration; and     -   receive uplink signals comprising a plurality of CSI reports         including an explicit CSI from the communication device 202 a,         202 b; and         a processor 200 a configured to construct a precoder matrix         applied on the antenna ports using the explicit CSI.         System

The present invention provides a wireless communication network, comprising at least one of the inventive communication devices 202 a, 202 b, and at least one of the inventive transmitters 200.

In accordance with embodiments, the communication device and the transmitter comprises one or more of a mobile terminal, or stationary terminal, or cellular IoT-UE, or an IoT device, or a ground based vehicle, or an aerial vehicle, or a drone, or a moving base station, or road side unit, or a building, or a macro cell base station, or a small cell base station, or a road side unit, or a UE, or a remote radio head, or an AMF, or a SMF, or a core network entity, or a network slice as in the NR or 5G core context, or any transmission/reception point (TRP) enabling an item or a device to communicate using the wireless communication network, the item or device being provided with network connectivity to communicate using the wireless communication network.

Methods

The present invention provides a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising:

receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, performing a high-order principal component analysis, HO-PCA, on the channel tensor, identifying a plurality of dominant principal components of the channel tensor, thereby obtaining a compressed channel tensor, and reporting the explicit CSI comprising the dominant principal components of the channel tensor from the communication device to the transmitter.

The present invention provides a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising:

receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, calculating a transformed channel tensor using the channel tensor, rewriting the transformed channel tensor to a transformed channel matrix, performing a standard principal component analysis, PCA, on the transformed channel matrix, identifying a plurality of dominant principal components of the transformed channel matrix, thereby obtaining a transformed/compressed channel matrix, and reporting the explicit CSI comprising the plurality of dominant principal components of the transformed/compressed channel tensor from the communication device to the transmitter.

The present invention provides a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising:

receiving, from a transmitter, a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and estimating the CSI using measurements on the downlink reference signals of the radio channel, constructing a frequency-domain channel matrix using the CSI estimate, performing a standard principal component analysis, PCA, on the channel matrix, identifying the dominant principal components of the channel matrix, the channel matrix comprising

-   -   a first set of r basis vectors contained in a matrix Ū=[u₁,u₂, .         . . ,u_(r)]∈         ^(N) ^(t) ^(N) ^(r) ^(×r);     -   a second set of r basis vectors contained in a matrix V=[v₁, v₂,         . . . , v_(r)]∈         ^(S×r); and     -   a third set of r coefficients contained in a diagonal matrix         Σ=diag(s₁, s₂, . . . , s_(r))∈         ^(r×r) with ordered singular values s_(i) (s₁≥s₂≥ . . . ≥s_(R))         on its diagonal;     -   calculating a reduced-sized delay-domain matrix {tilde over (V)}         from the frequency-domain matrix V, wherein the delay-domain         matrix, comprising r basis vectors, is given by         V≈F _(V) {tilde over (V)},     -   where         -   F_(V)∈             ^(S×L) contains L vectors of size S×1, selected from a             discrete Fourier transform, DFT, codebook Ω, the size of the             compressed delay-domain matrix {tilde over (V)}=[{tilde over             (v)}₁, {tilde over (v)}₂, . . . , {tilde over (v)}_(r)] is             given by L×r, and L is the number of delays, and         -   the oversampled codebook matrix is given by Ω=[d₀, d₁, . . .             , d_(SO) _(f) ⁻¹], where

${\left\lbrack {1\mspace{14mu} e^{\frac{{- j}\; 2\pi\; i}{O_{f}S}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\pi\;{i{({S - 1})}}}{O_{f}S}}} \right\rbrack^{T} \in {\mathbb{C}}^{S \times 1}},$ i∈{0, . . . , SO_(f)−1}, and O_(f)∈{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; and

-   -   reporting the explicit CSI containing the identified first set         of r basis vectors, the second reduced-sized delay-domain set of         r basis vectors, along with the L delays, represented by a set         of indices that correspond to the selected DFT-vectors in the         codebook Ω from the communication device to the transmitter.

The present invention provides a method for transmitting in a wireless communication system including a communication device a communication device of any one of the claims 1 to 14 and a transmitter, the method comprising:

transmitting, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS configuration, and downlink signals comprising the CSI-RS configuration;

receiving, at the transmitter, uplink signals comprising a plurality of CSI reports including an explicit CSI from the communication device;

constructing a precoder matrix for a precoder connected to an antenna array having a plurality of antennas;

applying the precoder matrix on antenna ports using the explicit CSI so as to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams.

Computer Program Product

The present invention provides a computer program product comprising instructions which, when the program is executed by a computer, causes the computer to carry out one or more methods in accordance with the present invention.

Thus, the present invention provides several low feedback overhead approaches for explicit CSI reporting based on channel transformations and compression techniques, and embodiments relate to wireless communications systems and, more specifically, to frequency-domain, delay-domain, time domain or mixed frequency/delay and time/Doppler-frequency domain explicit CSI feedback to represent a downlink channel between a gNB and a single UE in the form of a channel tensor or matrix, a beamformed channel tensor or matrix, a covariance channel tensor or matrix, dominant eigenvectors of a channel tensor or matrix or dominant eigenvectors of a beamformed channel tensor or matrix. In the following, several embodiments of low feedback overhead approaches for explicit (frequency-domain, delay-domain, time domain or mixed frequency/delay and time/Doppler-frequency domain) CSI reporting based on combinations of channel transformations and compression techniques will be described.

High Order PCA on Frequency-Domain Channel Tensor

In accordance with a first embodiment 1, a UE is configured to report “explicit CSI Type I” that represents a compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed channel covariance tensor over the configured subbands (SB), PRBs or subcarriers, according to the following sub-embodiments. Here, an SB corresponds to a set of consecutive PRBs. For example, for a bandwidth part of 10 MHz, 6 subbands each having 8 PRBs are configured.

The compressed CSI is based on a high-order principal component analysis (HO-PCA) of the channel tensor to exploit the correlations of the channel tensor in the space- and frequency-domains.

An illustration of this approach is shown in FIG. 4 . The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a three-dimensional (3D) frequency-domain channel tensor (a three-dimensional array)

∈

^(N) ^(r) ^(×N) ^(t) ^(×s) of dimension N_(r)×N_(t)×S, where S is the number of subbands, PRBs, or subcarriers (see FIG. 5 ). The definition of N_(t) and N_(r) is dependent on the configuration of the CSI type:

-   -   N_(t) is the number of transmit antenna ports 2N₁N₂ for CSI type         configuration “channel tensor”, N_(t)=2N₁N₂, and N_(r) is the         number of UE receive antenna ports;     -   N_(t) is the number of transmit antenna ports 2N₁N₂ at the gNB,         N_(t)=2N₁N₂, N_(r)=2N₁N₂, for CSI type configuration “channel         covariance tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of UE receive antenna ports for CSI type         configuration “beamformed-channel tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         beams 2U and N_(r) is the number of beamformed antenna         ports/beams N_(t)=2U beams 2U for CSI type configuration         “beamformed-channel covariance tensor”;

Then, the UE performs a HO-PCA on the channel tensor

, such that

is represented by

${\mathcal{H} = {\sum\limits_{i = 1}^{N_{r}}{\sum\limits_{j = 1}^{N_{t}}{\sum\limits_{k = 1}^{S}{s_{ijk}\left( {u_{R,i} \circ u_{T,j} \circ u_{S,k}} \right)}}}}},$ where

-   -   U_(R)=[U_(R,1), . . . ,U_(R,N) _(r) ]∈         ^(N) ^(r) ^(×N) ^(r) is matrix containing the higher order         singular vectors with respect to the first dimension of the         channel tensor         ;     -   U_(T)=[u_(T,1), . . . ,u_(T,N) _(t) ]∈         ^(N) ^(t) ^(×N) ^(t) is matrix containing the higher order         singular vectors with respect to the second dimension of the         channel tensor         ;     -   U_(S)=[u_(S,1), . . . , u_(S,R)]∈         ^(S×R) is a matrix containing the higher order singular vectors         with respect to the frequency dimension (third dimension of the         channel tensor         ) with R=min(S, N_(t)N_(r));     -   S_(ijk) are the higher order singular values, sorted as         s_(ijk)≥s_(i′j′k′) with i′≤i, j′≤j, k′≤k.

Moreover, the º symbol represents the outer product operator which is the generalization of the outer product of two vectors (giving rise to a matrix that contains the pairwise products of all the elements of the two vectors) to multi-way matrices/tensors. The outer product of an R-way tensor

(i.e., a matrix that is indexed by R indices and a P-way tensor

is a (R+P)-way tensor

containing all pairwise products of all the elements of

and

. Note that vectors and matrices may be seen as 1-way and 2-way tensors respectively. Therefore, the outer product between two vectors is a matrix, the outer product between a vector and a matrix is a 3-way tensor, and so on.

To reduce the number of channel coefficients, the channel tensor i is approximated by (r₁,r₂,r₃), (1≤r₁≤N_(r), 1≤r₂≤N_(t), 1≤r₃≤S) dominant principal components with respect to the first, second and third dimensions and the corresponding left, right and lateral singular matrices, respectively. The compressed explicit frequency-domain channel tensor (explicit CSI) is given by

$\mathcal{H}_{c} = {\sum\limits_{i = 1}^{r_{1}}{\sum\limits_{j = 1}^{r_{2}}{\sum\limits_{k = 1}^{r_{3}}{{s_{ijk}\left( {u_{R,i} \circ u_{T,j} \circ u_{S,k}} \right)}.}}}}$ where Ū _(R)=[u_(R,1) , . . . ,u _(R,r) ₁ ]∈

^(N) ^(r) ^(×r) ¹ , Ū _(T)=[u _(T,1) , . . . ,u _(T,r) ₂ ]∈

^(N) ^(t) ^(×r) ² , Ū _(S)=[u _(S,1) , . . . ,u _(S,r) ₃ ]∈

^(S×r) ³ .

To report the compressed frequency-domain channel tensor (explicit CSI) from the UE to the gNB, the UE quantizes the coefficients of the vectors u_(R,i), u_(T,j), and u_(S,k) and the HO singular values s_(ijk) using a codebook approach.

The gNB reconstructs the compressed channel tensor as

$\mathcal{H}_{c} = {\sum\limits_{i = 1}^{r_{1}}{\sum\limits_{j = 1}^{r_{2}}{\sum\limits_{k = 1}^{r_{3}}{{s_{ijk}\left( {u_{R,i} \circ u_{T,j} \circ u_{S,k}} \right)}.}}}}$

The number of complex coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by N_(r)r₁+N_(t)r₂+Sr₃ for the higher-order singular vectors and number of real coefficients that need to be quantized for the frequency-domain HO-PCA approach is r₁r₂r₃, for the higher-order singular values, respectively. In comparison, for the standard (non-high order) PCA, it is needed to quantize N_(r)N_(t)r+Sr+r values for the singular vectors and the singular values with r=min(N_(r)N_(t), S). For small values of (r₁, r₂, r₃) (low rank approximation of the channel tensor), the compression achieved by the HO-PCA is higher than the compression achieved by the standard non-HO PCA approach.

In one method, the values of (r₁, r₂, r₃), representing the number of dominant principal components with respect to the first, second and third dimension of the channel tensor, respectively, are configured via higher layer (e.g., RRC, or MAC-CE) signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r₁,r₂,r₃) as a part of the CSI report, or they are known at the UE.

In accordance with a sub-embodiment 1-1 of the first embodiment 1, as illustrated in FIG. 6 , a UE is configured to report “explicit CSI Type I” with “delay-domain CSI for the higher-order singular-value matrix Us” In this configuration, the UE calculates an approximated reduced-sized (compressed) delay-domain higher-order singular-matrix Ũ_(S) from the frequency-domain higher-order singular-matrix Ū_(S). The delay-domain higher-order singular-matrix is given by Ū _(S) ≈F _(S) Ũ _(S), where F_(S)∈

^(S×L) is non-square, or square matrix of size S×L consisting of L discrete Fourier transform (DFT) vectors. The size of the compressed delay-domain matrix Ũ_(S) is given by L×r₃. A compression is achieved when L<S.

The DFT vectors in F_(S) are selected from a oversampled DFT-codebook matrix Ω=[d₀, d₁, . . . , d_(SO) _(f) ⁻¹], where

${d_{i} = {\left\lbrack {1\mspace{14mu} e^{\frac{{- j}\; 2\pi\; i}{O_{f}S}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\pi\;{i{({S - 1})}}}{O_{f}S}}} \right\rbrack^{T} \in {\mathbb{C}}^{S \times 1}}},$ i=0, . . . , O_(f)S−1. Here, 0_(f){1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix. The indices of the vectors in F_(S) selected from the codebook Ω are stored in a set

=(i₁, i₂, . . . , i_(L)).

The UE quantizes the coefficients of the vectors in Ū_(R)=[u_(R,1), . . . ,u_(R,r) ₁ ], Ū_(T)=[u_(T,1), . . . ,u_(T,r) ₂ ], and Ũ_(S)=[ũ_(S,1), . . . , ũ_(S,r) ₃ ]∈

^(L×r) ³ and the HO singular values s_(ijk) using a codebook approach, and reports them along with the L delays, represented by a set of indices

that correspond to the selected DFT vectors in the codebook Ω, to the gNB. The gNB reconstructs the frequency-domain channel tensor

_(c) according to embodiment 1, where Ū_(S) is calculated as Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ]=F_(S)Ũ_(S).

In one method, the number of delays L is configured via higher layer (e.g., RRC, or MAC) signaling from the gNB to the UE. In another method, the UE reports the preferred value of L as a part of the CSI report, or it is known at the UE.

The oversampled factor O_(f) of the DFT codebook matrix is configured via higher layer (e.g., RRC, or MAC) or physical layer (via the downlink control indicator (DCI)) signaling from the gNB to the UE, or it is known at the UE.

Standard (Non-High-Order) PCA of Frequency-Domain Channel Matrix with Delay-Domain Compression

In accordance with a second embodiment 2, a UE is configured to report “explicit CS Type II” that represents a compressed form of a channel matrix, or a beam-formed channel matrix, or a channel covariance matrix, or beam-formed channel covariance matrix over the configured subbands (SB), PRBs or subcarriers, according to the following sub-embodiments.

The compressed CSI performs a standard non-high-order principal component analysis (non-HO-PCA) on a channel matrix combined with a delay-domain transformation and compression of the channel matrix.

An illustration of this approach is shown in FIG. 7 . The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 2D frequency-domain channel matrix H∈

^(N) ^(t) ^(N) ^(r) ^(×S) of dimension N_(t)N_(r)×S, where S is the number of subbands, PRBs, or subcarriers (see FIG. 8 ). The definition of N_(t) and N_(r) is dependent on the configuration of the CSI type:

-   -   N_(t) is the number of transmit antenna ports 2N₁N₂ for CSI type         configuration “channel matrix”, N_(t)=2N₁N₂, and N_(r) is the         number of UE receive antenna ports;     -   N_(t) is the number of transmit antenna ports 2N₁N₂ at the gNB,         N_(t)=2N₁N₂, N_(r)=2N₁N₂, for CSI type configuration “channel         covariance matrix”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of UE receive antenna ports for CSI type         configuration “beamformed-channel matrix”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of beamformed antenna ports/beams         N_(r)=2U for CSI type configuration “beamformed-channel         covariance matrix”;

The UE applies a standard PCA-decomposition to the frequency-domain channel matrix H, represented by

${H = {{U\Sigma V^{H}} = {\sum\limits_{i = 1}^{R}{s_{i}u_{i}v_{i}^{H}}}}},$ where

-   -   U=[u₁, u₂, . . . , u_(R)] is the N_(t)N_(r)×R left-singular         matrix;     -   V=[v₁, v₂, . . . , v_(R)] is the S×R right-singular matrix;     -   Σ is a R×R diagonal matrix with ordered singular values s_(i)         (s₁≥s₂≥ . . . ≥S_(R)) on its main diagonal, and R=min(S,         N_(t)N_(t)).

To reduce the number of channel coefficients, the channel matrix H is approximated by r, 1≤r≤R dominant principal components. The “compressed” channel matrix H_(c) is given by H _(c) =ŪΣ V ^(H) where Ū=[u₁, u₂, . . . , u_(r)], V=[v₁, v₂, . . . v_(r)], and Σ=diag(s₁, s₂, . . . , s_(r)).

Furthermore, the UE calculates from the frequency-domain right-singular matrix V the corresponding “compressed” delay-domain right singular matrix. The delay-domain right-singular matrix is approximated by V≈F _(V) {tilde over (V)}, where F_(V)∈

^(S×L) is a squared, or non-squared DFT matrix of size S×L. The size of the compressed delay-domain left-singular matrix is given by L×r. A compression is achieved for L<S.

The columns of the transformation/compression matrix F_(V) are selected from a DFT codebook matrix (Ω) of dimension S×SO_(f), where O_(f) denotes the oversampling factor of the DFT codebook-matrix. The indices of the selected vectors in F_(V) from the codebook are stored in a set

=(i₁, i₂, . . . , i_(L)).

The UE quantizes the frequency-domain left singular-matrix Ū, the delay-domain right singular-matrix {tilde over (V)} and the singular values s₁, s₂, . . . , s_(r) using a codebook approach, and then reports them along with the L delays, represented by the index set

to the gNB.

The gNB reconstructs the explicit CSI as H _(c) =Ū Σ(F{tilde over (V)})^(H).

In one method, the number of delays L is configured via higher layer (e.g., RRC, or MAC) signaling, or physical layer (via the downlink control indicator (DCI)) signaling from the gNB to the UE. In another method, the UE reports the preferred value of L as a part of the CSI report, or it is known at the UE.

In one method, the value of r, representing the number of dominant principal components of the channel matrix is configured via higher layer (e.g., RRC, or MAC-CE) signaling from the gNB to the UE. In another method, the UE reports the preferred value of r as a part of the CSI report, or it is known at the UE.

The oversampled factor O_(f) of the DFT codebook matrix is configured via higher layer (e.g., RRC, or MAC) or via DCI physical signaling from the gNB to the UE, or it is known at the UE.

Transformation/Compression of Channel Tensor in Combination with HO-PCA

In accordance with a third embodiment 3, a UE is configured to report “explicit CSI Type III” that represents a transformed and compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed covariance tensor over the configured subbands (SB), PRBs or subcarriers with respect to the space, frequency, or space and frequency dimension of the channel tensor. The CSI combines channel tensor transformation with data compression by exploiting the sparse representation in the delay domain and the correlations of the channel coefficients in the spatial and frequency/delay domains.

An illustration of this approach is shown in FIG. 9 . The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 3D frequency-domain channel tensor

∈

^(N) ^(r) ^(×N) ^(t) ^(×S) of dimension N_(r)×N_(t)×S, where S is the number of subbands, PRBs, or subcarriers. The definition of N_(t) and N_(r) is dependent on the configuration of the CSI type:

-   -   N_(t) is the number of transmit antenna ports 2N₁N₂ for CSI type         configuration “channel tensor”, N_(t)=2N₁N₂, and N_(r) is the         number of UE receive antenna ports;     -   N_(t) is the number of transmit antenna ports 2N₁N₂ at the gNB,         N_(t)=2N₁N₂, N_(r)=2N₁N₂, for CSI type configuration “channel         covariance tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of UE receive antenna ports for CSI type         configuration “beamformed-channel tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of beamformed antenna ports/beams         N_(r)=2U for CSI type configuration “beamformed-channel         covariance tensor”;

After the construction of the frequency-domain channel tensor

, a one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D) transformation of the channel tensor is applied with respect to the space, frequency, or frequency and space dimensions of the channel tensor. The aim of the transformation is to obtain a sparse or nearly-sparse representation of the channel tensor in one, two, or three dimensions. After the transformation and compression step, the size of the channel tensor is reduced and a compression with respect to one, two or three dimensions of the channel tensor is achieved. For example, a transformation/compression with respect to all three dimensions of the channel tensor

is represented by a (column-wise) Kronecker product as

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{({1,2,3})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2,3})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,n_{r},n_{t},s} \otimes b_{2,n_{r},n_{t},s} \otimes b_{1,n_{r},n_{t},s}}}}}}}},}$ where

-   -   b_(1,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         N_(r)×1 with respect to the first dimension of the channel         tensor         , selected from a codebook matrix Ω₁;     -   b_(2,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         N_(t)×1 with respect to the second dimension of the channel         tensor         , selected from a codebook matrix Ω₂;     -   b_(3,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         S×1 with respect to the third dimension of the channel tensor         , selected from a codebook matrix Ω₃;     -   γ_(n) _(r) _(,n) _(t) _(,s) is the transformed/compressed         channel coefficient associated with the vectors b_(1,n) _(r)         _(,n) _(t) _(,s), b_(2,n) _(r) _(,n) _(t) _(,s), and b_(3,n)         _(r) _(,n) _(t) _(,s), and     -   N′_(r), N′_(t), and S′ represents the value of the first, second         and third dimension of the transformed/compressed channel tensor         , respectively.

The transformed/compressed channel coefficients γ_(n) _(r) _(,n) _(t) _(,s) are used to form the transformed/compressed channel tensor

of dimension N′_(r)×N′_(t)×S′, where N′_(r)≤N_(r), N′_(t)≤N_(t), S′≤S.

For example, a transformation/compression with respect to the two space dimensions of the channel tensor

is represented by

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,s} \otimes b_{2,n_{r},n_{t},s} \otimes b_{1,n_{r},n_{t},s}}}}}}}},}$ where b_(3,s) is a vector of all zeros with the s-th element being one, b_(2,n) _(r) _(,n) _(t) _(,s) is a vector of size N_(t)×1 with respect to the second dimension of the channel tensor

, selected from a codebook matrix Ω₂, b_(1,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size N_(r)×1 with respect to the first dimension of the channel tensor

, selected from a codebook matrix Ω₁ and N′_(r)≤N_(r)N′_(t)≤N_(t), S′=S.

For example, a transformation/compression with respect to the frequency dimension of the channel tensor

is represented by

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,n_{r},n_{t},s} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}$ where b_(2,n) _(r) is a vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one, b_(3,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size S×1 with respect to the third dimension of the channel tensor

, selected from a codebook matrix Ω₃, and N′_(r)=N_(r), N′_(t)=N_(t) and S′≤S.

The indices of the selected vectors b_(1,n) _(r) _(,n) _(t) _(,s), b_(2,n) _(r) _(,n) _(t) _(,s) and b_(3,n) _(r) _(,n) _(t) _(,s) from the codebook matrices Ω_(n), n=1,2,3 are stored in a set

of g-tuples, where g refers to the number of transformed dimensions.

For example, for g=1, the set

is represented by

={i₁, i₂, i_(T)}, where T denotes the number of selected vectors with respect to the transformed/compressed dimension of the channel tensor

. For example, in the case of a transformation/compression with respect to the frequency dimension, T=S′.

For example, for g=3, the set

is represented by 3-tuples (i_(1,n) _(r) ,i_(2,n) _(t) ,i_(3,s)), where i_(1,n) _(r) is the index associated with vectors b_(1,n) _(r) _(,n) _(t) _(,s), i_(2,n) _(t) , is the index associated with vector b_(2,n) _(r) _(,n) _(t) _(,s), and i_(3,s) is the index associated with vector b_(3,n) _(r) _(,n) _(t) _(,s). The set

is given by

={(i _(1,0) ,i _(2,0) ,i _(3,0)), . . . ,(i _(1,N′) _(r) ⁻¹ ,i _(2,N′) _(t) ⁻¹ ,i _(3,S′−1))}

In one method, the size of the set

is configured via higher layer (e.g., RRC, or MAC) or physical layer signaling from the gNB to the UE. In another method, the UE reports the preferred size of the set as a part of the CSI report or it is known at the UE.

The codebook matrices Ω_(n) are given by matrices Ω_(n)=[d_(n,0), d_(n,1), . . . , d_(n,TO) _(n) ⁻¹], where the parameter O_(n) denotes the oversampling factor with respect to the n-th dimension where T=N_(r) for n=1, T=N_(t) for n=2, and T=S for n=3.

The oversampled factors O_(f,n) of the codebook matrices are configured via higher layer (e.g., RRC, or MAC) or via DCI physical layer signaling from the gNB to the UE, or they are known at the UE.

As an example, the selection of the transformation/compression vectors and transformed/compressed channel coefficients for a transformation/compression with respect to the second and third dimension, can be calculated by min∥vec(

)−vec(

_((2,3))(

))∥₂ ². The optimization problem may be solved by standard algorithms such as orthogonal matching pursuit. As a result, the indices of the vectors in the transformation matrices selected from the codebooks and the transformed channel coefficients associated with each domain are known.

After the channel transformation/compression, the UE performs a HO-PCA on the transformed/compressed channel tensor

, such that

is represented by

${\hat{\mathcal{H}} = {\sum\limits_{i = 1}^{N_{r}^{\prime}}{\sum\limits_{j = 1}^{N_{t}^{\prime}}{\sum\limits_{k = 1}^{S^{\prime}}{s_{ijk}\left( {u_{R,i} \circ u_{T,j} \circ u_{S,k}} \right)}}}}},$ where

-   -   U_(R)=[u_(R,1), . . . , u_(R,N′) _(r) ]∈         ^(N′) ^(r) ^(×N′) ^(r) is a matrix containing the high-order         singular vectors with respect to the first dimension of the         transformed/compressed channel tensor         ;     -   U_(T)=[u_(T,1), . . . , U_(T,N′) _(t) ]∈         ^(N′) ^(t) ^(×N′) ^(t) is matrix containing the high-order         singular vectors with respect to the second dimension of the         transformed/compressed channel tensor         ;     -   U_(S)=[u_(S,1), . . . , U_(S,R)]∈         ^(S′×R) is a matrix containing the high-order singular vectors         with respect to the third dimension of the channel tensor         with R=min(S′,N′_(t)N′_(r))     -   S_(ijk) are the high-order singular values, sorted as         s_(ijk)≥s_(i′j′k′), with i′≤i, j′≤j, k′≤k.

To further compress the number of channel coefficients, the transformed/compressed channel tensor

is approximated by (r₁,r₂,r₃), (1≤r₁≤N′_(r), 1≤r₂≤N′_(t), 1≤r₃≤S′) dominant principal components with respect to the first, second and third dimensions and the corresponding left, right and lateral singular matrices. The transformed/compressed explicit channel tensor (explicit CSI) is then given by

${\hat{\mathcal{H}}}_{c} = {\sum\limits_{i = 1}^{r_{1}}{\sum\limits_{j = 1}^{r_{2}}{\sum\limits_{k = 1}^{r_{3}}{{s_{ijk}\left( {u_{R,i} \circ u_{T,j} \circ u_{S,k}} \right)}.}}}}$ where Ū _(R)=[u _(R,1) , . . . ,u _(R,r) ₁ ]∈

^(N′) ^(r) ^(×r) ¹ , Ū _(T)=[u _(T,1) , . . . ,u _(T,r) ₁ ]∈

^(N′) ^(t) ^(×r) ² , Ū _(S)=[u _(S,1) , . . . ,u _(S,r) ₃ ]∈

^(S′×r) ³ .

The UE quantizes the coefficients of the vectors U_(R,i), U_(T,j), U_(S,k) and the singular values s_(ijkl) using a codebook approach. The quantized vectors u_(R,i), U_(T,j), U_(S,k) and the quantized singular values s_(ijk) along with the index set

are reported to the gNB.

The gNB reconstructs first the transformed/compressed channel tensor,

as

${\hat{\mathcal{H}}}_{c} = {\sum\limits_{i = 1}^{r_{1}}{\sum\limits_{j = 1}^{r_{2}}{\sum\limits_{k = 1}^{r_{3}}{{s_{ijk}\left( {u_{R,i} \circ u_{T,j} \circ u_{S,k}} \right)}.}}}}$

Then, based on the transformed/compressed channel tensor

_(c) and the signaled index set

, the frequency-domain channel tensor

is reconstructed as vec(

)≈(

_((1,2,3))(

_(c))(three-dimensional transformation/compression); vec(

)≈

_((1,2))(

_(c))(two-dimensional transformation/compression); vec(

)≈

₍₃₎(

_(c))(one-dimensional transformation/compression).

The number of coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by N′_(r)r₁+N′_(t)r₂+S′r₃+r₁r₂r₃, for the higher-order singular vectors and the higher-order singular values.

In one method, the values of (r₁,r₂,r₃), representing the number of dominant principal components with respect to the first, second and third dimension of the transformed/compressed channel tensor

_(c), respectively, are configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r₁,r₂,r₃) as a part of the CSI report or they are known at the UE.

In accordance with a sub-embodiment 3 1 of the third embodiment 3, the 3D transformation/compression function

_((1,2,3))(

) is given by a two-dimensional Discrete Cosine transformation (2D-DCT) with respect to the space dimensions and a 1D-DFT transformation with respect to the frequency dimension of the channel tensor. The codebook matrices Ω_(n), n=1,2 are given by oversampled discrete cosine transform (DCT) matrices. The codebook matrix Ω₃ is given by an oversampled DFT matrix.

In accordance with a sub-embodiment 3 2 of the third embodiment 3, the 3D transformation/compression function

_((1,2,3))(

) is given by a 3D-DFT transformation and the codebook matrices Ω_(n), n=1,2,3 are given by oversampled DFT matrices.

In accordance with a sub-embodiment 3 3 of the third embodiment 3, the 2D transformation/compression unction

_((1,2))(

) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices.

In accordance with a sub-embodiment 3 4 of the third embodiment 3, the 1D transformation/compression function

₍₃₎(

) is given by a 1D-DFT transformation and the codebook matrix Ω₃ is given by an oversampled DFT matrix.

Transformation/Compression of Channel Matrix in Combination with Standard Non-HO-PCA

In accordance with a fourth embodiment 4, a UE is configured to report “explicit CSI Type IV” that represents a transformed and compressed form of a channel matrix, or a beam-formed channel tensor, or a channel covariance tensor over the configured subbands (SB), PRBs or subcarriers with respect to the space, frequency, or space and frequency dimension of the channel matrix. The CSI combines channel tensor transformation with data compression by exploiting the sparse representation in the delay domain and the correlations of the channel coefficients in the spatial and frequency/delay domains.

An illustration of this approach is shown in FIG. 10 . The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a three-dimensional (3D) frequency-domain channel tensor (a three-dimensional array)

∈

^(N) ^(r) ^(N) ^(t) ^(×S) of dimension N_(r)×N_(t)×S, where S is the number of subbands, PRBs, or subcarriers. The definition of N_(t) and N_(r) is dependent on the configuration of the CSI type:

-   -   N_(t) is the number of transmit antenna ports 2N₁N₂ for CSI type         configuration “channel tensor”, N_(t)=2N₁N₂, and N_(r) is the         number of UE receive antenna ports;     -   N_(t) is the number of transmit antenna ports 2N₁N₂ at the gNB,         N_(t)=2N₁N₂, N_(r)=2N₁N₂, for CSI type configuration “channel         covariance tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of UE receive antenna ports for CSI type         configuration “beamformed-channel tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of beamformed antenna ports/beams         N_(r)=2U for CSI type configuration “beamformed-channel         covariance tensor”;

After the construction of the frequency-domain channel tensor

, a one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D) transformation and compression of the channel tensor is applied with respect to the space, frequency, or frequency and space dimensions of the channel tensor. The aim of the transformation is to obtain a sparse or nearly-sparse representation of the channel tensor in one, two, or three dimensions and to extract the dominant coefficients having the highest energy. After the transformation/compression step, the size of the channel tensor is reduced and a compression with respect to one, two or three dimensions of the channel tensor is achieved.

For example, a transformation/compression with respect to all three dimensions of the channel tensor

is represented by a (column-wise) Kronecker product as

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{({1,2,3})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2,3})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,n_{r},n_{t},s} \otimes b_{2,n_{r},n_{t},s} \otimes b_{1,n_{r},n_{t},s}}}}}}}},}$ where

-   -   b_(1,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         N_(r)×1 with respect to the first dimension of the channel         tensor         , selected from a codebook matrix Ω₁;     -   b_(2,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         N_(t)×1 with respect to the second dimension of the channel         tensor         , selected from a codebook matrix Ω₂;     -   b_(3,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size         S×1 with respect to the third dimension of the channel tensor         , selected from a codebook matrix Ω₃;     -   γ_(n) _(r) _(,n) _(t) _(,s) is the transformed/compressed         channel coefficient associated with the vectors b_(1,n) _(r)         _(,n) _(t) _(,s), b_(2,n) _(r) _(,n) _(t) _(,s), and b_(3,n)         _(r) _(,n) _(t) _(,s), and     -   N′_(r), N′_(t), and S′ represents the value of the first, second         and third dimension of the transformed/compressed channel tensor         , respectively.

The transformed/compressed channel coefficients γ_(n) _(r) _(,n) _(t) _(,s) are used to form the transformed/compressed channel tensor

of dimension N′_(r)×N′_(t)×S′.

For example, a transformation/compression with respect to the two space dimensions of the channel tensor

is represented by

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,s} \otimes b_{2,n_{r},n_{t},s} \otimes b_{1,n_{r},n_{t},s}}}}}}}},}$ where b₃,s is a vector of all zeros with the s-th element being one, S′=S, b_(2,n) _(r) _(,n) _(t) _(,s) is a vector of size N_(t)×1 with respect to the second dimension of the channel tensor

, selected from a codebook matrix Ω₂, b_(1,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size N_(r)×1 with respect to the first dimension of the channel tensor

, selected from a codebook matrix Ω₁, N′_(r)≤N_(r),N′_(t)≤N_(t) and S′=S.

For example, a transformation/compression with respect to the frequency dimension of the channel tensor

is represented by

$\mspace{20mu}{{\mathcal{H} = {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,n_{r},n_{t},s} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}$ where b_(2,n) _(r) is a vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one, b_(3,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size S×1 with respect to the third dimension of the channel tensor

, selected from a codebook matrix Ω₃, N′_(r)=N_(r), N′_(t)=N_(t) and S′≤S.

As an example, the selection of the transformation/compression vectors and transformed/compressed channel coefficients for a transformation/compression with respect to the second and third dimension, can be calculated by min∥vec(

)−vec(

_((2,3))(

))∥₂ ². The optimization problem may be solved by standard algorithms such as orthogonal matching pursuit. As a result, the indices of the vectors in the transformation matrices selected from the codebooks and the transformed channel coefficients associated with each domain are known.

The indices of the selected vectors b_(1,n) _(r) _(,n) _(t) _(,s), b_(2,n) _(r) _(,n) _(t) _(,s) and b_(3,n) _(r) _(,n) _(t) _(,s) from the codebook matrices Ω_(n), n=1,2,3 are stored in a set

of g-tuples, where g refers to the number of transformed dimensions.

For example, for g=1, the set

is represented by

={i₁, i₂, . . . , i_(T)}, where T denotes the number of selected vectors with respect to the transformed/compressed dimension of the channel tensor

. For example, in the case of a transformation/compression with respect to the frequency dimension, T=S′.

For example, for g=3, the set

is represented by 3-tuples (i_(1,n) _(r) , i_(2,n) _(t) , i_(3,s)), where i_(1,n) _(r) is the index associated with vectors b_(1,n) _(r) _(,n) _(t) _(,s), i_(2,n) _(t) , is the index associated with vector b_(2,n) _(r) _(,n) _(t) _(,s), and i_(3,s) is the index associated with vector b_(3,n) _(r) _(,n) _(t) _(,s). The set

is given by

={(i _(1,0) ,i _(2,0) ,i _(3,0)), . . . ,(i _(1,N′) _(r) ⁻¹ ,i _(2,N′) _(t) ⁻¹ ,i _(3,S′−1))} In one method, the size of the set

is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred size of the set as a part of the CSI report or it is known at the UE.

The codebook matrices n are given by matrices Ω_(n)=[d_(n,0), d_(n,1), . . . , d_(n,TO) _(n) ⁻¹], where the parameter O_(f,n) denotes the oversampling factor with respect to the n-th dimension where T=N_(r) for n=1, T=N_(t) for n=2, and T=S for n=3.

The oversampled factors O_(f,n) of the codebook matrices are configured via higher layer or via DCI physical layer signaling from the gNB to the UE, or they are known at the UE.

After the channel transformation/compression, the UE rewrites the transformed/compressed channel tensor to a transformed/compressed channel matrix Ĥ, and applies a standard PCA decomposition, represented by

${\hat{H} = {{U\Sigma V^{H}} = {\sum\limits_{i = 1}^{R}{s_{i}u_{i}v_{i}^{H}}}}},$ where

-   -   U=[u₁,u₂, . . . ,u_(R)] is the N′_(t)N′_(r)×R left-singular         matrix;     -   V=[v₁, v₂, . . . ,v_(R)] is the S′×R right-singular matrix;     -   Σ is a R×R diagonal matrix with ordered singular values s_(i)         (s₁≥s₂≥ . . . ≥S_(R)) on its main diagonal, and R=min(S′,         N′_(t)N′_(r)).

The transformed/compressed channel matrix Ĥ_(c) is then constructed using r, 1≤r≤R dominant principal components as Ĥ _(c) =ŪΣ V ^(H), where Ū=[u₁, u₂, . . . , u_(r)] and Σ=diag(s₁, s₂, . . . , s_(r)), and V=[v₁, v₂, . . . , v_(r)].

The UE quantizes U, V and the singular values s₁, s₂, . . . , s_(r) using a codebook approach, and then reports them along with the index set

, to the gNB.

The gNB reconstructs first the transformed/compressed channel matrix Ĥ_(c) as Ĥ _(c) =ŪΣ V ^(H).

Then, based on the transformed/compressed channel matrix, the gNB constructs the transformed/compressed channel tensor

_(c). Using the signaled index set

, the frequency-domain channel tensor

is reconstructed as vec(

)≈

_((1,2,3))(

_(c))(three-dimensional transformation/compression); vec(

)≈

_((1,2))(

_(c))(two-dimensional transformation/compression); vec(

)≈

₍₃₎(

_(c))(one-dimensional transformation/compression).

In one method, the value of r representing the number of dominant principal components of the transformed/compressed channel matrix Ĥ_(c) is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r₁, r₂, r₃) as a part of the CSI report or they are known at the UE.

In accordance with a sub-embodiment 4 1 of the fourth embodiment 4, the 3D transformation/compression function

_((1,2,3))(

) is given by a 2D-DCT transformation with respect of the space dimensions and a 1D-DFT transformation with respect to the frequency dimension of the channel tensor. The codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices. The codebook matrix Ω₃ is given by an oversampled DFT matrix.

In accordance with a sub-embodiment 4 2 of the fourth embodiment 4, the 3D transformation/compression function

_((1,2,3))(

) is given by a 3D DFT transformation and the codebook matrices Ω_(n), n=1,2,3 are given by oversampled DFT matrices.

In accordance with a sub-embodiment 4 3 of the fourth embodiment 4, the 2D transformation/compression function

_((1,2))(

) is given by a 2D Discrete Cosine transformation (DCT) and the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices.

In accordance with a sub-embodiment 4 4 of the fourth embodiment 4, the 1D transformation/compression function

₍₃₎(

) is given by a 1D DFT transformation and the codebook matrix Ω₃ is given by an oversampled DFT matrix.

Extension to Doppler Frequency Domain: High Order PCA on Four-Dimensional Frequency-Domain Channel Tensor

In accordance with a fifth embodiment 5, a UE is configured to report “explicit CSI Type V” that represents a compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed channel covariance tensor over the configured subbands (SB), PRBs or subcarriers according to the following sub-embodiments.

The compressed CSI is based on a high-order principal component analysis (HO-PCA) of the four-dimensional channel tensor to exploit the correlations of the channel tensor in the space-, frequency- and time/Doppler-frequency domains.

An illustration of this approach is shown in FIG. 11 . The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 4D frequency-domain channel

∈

^(N) ^(r) ^(×N) ^(t) ^(×S×D) of dimension N_(r)×N_(t)×S×D, where S is the number of subbands, PRBs, or subcarriers and D is number of snapshots of the channel measured at D consecutive time instants/slots. The definition of N_(t) and N_(r) is dependent on the configuration of the CSI type:

-   -   N_(t) is the number of transmit antenna ports 2N₁N₂ for CSI type         configuration “channel tensor”, N_(t)=2N₁N₂, and N_(r) is the         number of UE receive antenna ports;     -   N_(t) is the number of transmit antenna ports 2N₁N₂ at the gNB,         N_(t)=2N₁N₂, N_(r)=2N₁N₂ for CSI type configuration “channel         covariance tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of UE receive antenna ports for CSI type         configuration “beamformed-channel tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of beamformed antenna ports/beams         N_(r)=2U for CSI type configuration “beamformed-channel         covariance tensor”;

Then, the UE performs a HO-PCA on the four-dimensional channel tensor

, such that

is represented by

$\mathcal{H} = {\sum\limits_{i = 1}^{N_{r}}\;{\sum\limits_{j = 1}^{N_{t}}\;{\sum\limits_{k = 1}^{S}\;{\sum\limits_{l = 1}^{D}\;{{S_{ijkl}\left( {u_{R,i} \cdot u_{T,j} \cdot u_{S,k} \cdot u_{D,l}} \right)}.}}}}}$ where

-   -   U_(R)=[U_(R,1), . . . ,u_(R,N) _(r) ]∈         ^(N) ^(r) ^(×N) ^(r) is matrix containing the higher order         singular vectors with respect to the receive antennas (first         dimension of the channel tensor         );     -   U_(T)=[u_(T,1), . . . , u_(T,N) _(t) ]∈         ^(N) ^(t) ^(×N) ^(t) is matrix containing the higher order         singular vectors with respect to the transmit antennas (second         dimension of the channel tensor         );     -   U_(S)=[u_(S,1), . . . , u_(S,S)]∈         ^(S×S) is a matrix containing the higher order singular vectors         with respect to the frequency dimension (third dimension of the         channel tensor         );     -   U_(D)=[u_(D,1), . . . , u_(D,R)]∈         ^(D×R) is a matrix containing the higher order singular vectors         with respect to the time/channel snapshot dimension (fourth         dimension of the channel tensor         ), where R is the rank of the channel tensor given by         R=min{N_(r)N_(t)S, D};     -   s_(ijkl) are the higher order singular values, sorted as         s_(ijkl)≥s_(i′j′k′l′) with i′≤i, j′≤j, k′≤k, l′≤l.

To reduce the number of channel coefficients, the channel tensor

is approximated by (r₁,r₂,r₃,r₄), (1≤r₁≤N_(r), 1≤r₂≤N_(t), 1≤r₃≤S, 1≤r₄≤D) dominant principal components with respect to the first, second, third and fourth dimensions and the corresponding 1-mode (left), 2-mode (right) and 3-mode (lateral) and 4-mode singular matrices. The compressed explicit frequency-domain channel tensor (explicit CSI) is given by

_(c)=Σ₁₌₁ ^(r) ^(i) Σ_(j=1) ^(r) ² Σ_(k=1) ^(r) ³ Σ_(i=1) ^(r) ⁴ s _(ijkl)(u _(R,i) ºu _(T,j) ºu _(S,k) ºu _(D,l)), where Ū _(R)=[u _(R,1) , . . . ,u _(R,r) ₁ ]∈

^(N) ^(r) ^(×r) ¹ , Ū _(T)=[u _(T,1) , . . . ,u _(T,r) ₂ ]∈

^(N) ^(t) ^(×r) ² , Ū _(S)=[u _(S,1) , . . . ,u _(S,r) ₃ ]∈

^(S×r) ³ , Ū _(D)=[u _(D,1) , . . . ,u _(D,r) ₄ ]∈

^(D×r) ⁴ .

To report the compressed frequency-domain channel tensor (explicit CSI) from the UE to the gNB, the UE quantizes the coefficients of the vectors u_(R,i), u_(T,j), U_(S,k), U_(D,l) and the singular values s_(ijkl) using a codebook approach.

The gNB reconstructs the compressed channel tensor as

$\mathcal{H}_{c} = {\sum\limits_{i = 1}^{r_{1}}\;{\sum\limits_{j = 1}^{r_{2}}\;{\sum\limits_{k = 1}^{r_{3}}\;{\sum\limits_{l = 1}^{r_{4}}\;{{s_{ijkl}\left( {u_{R,i} \cdot u_{T,j} \cdot u_{S,k} \cdot u_{D,l}} \right)}.}}}}}$

The number of complex coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by N_(r)r₁+N_(t)r₂+Sr₃+Dr₄ for the higher-order singular vectors, and the number of real coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by r₁r₂r₃r₄ for the higher-order singular values, respectively. In comparison, for the standard (non-high order) PCA (see FIG. 12 ), it is needed to quantize N_(r)N_(t)r+SDr+r values for the singular vectors and the singular values with r=min(N_(r)N_(t), SD). For small values of (r₁, r₂, r₃, r₄) (low rank approximation of the channel tensor), the compression achieved by the HO-PCA is higher than the compression achieved by the standard non-HO PCA approach.

In one method, the values of (r₁, r₂, r₃, r₄), representing the number of dominant principal components with respect to the first, second, third and fourth dimension of the channel tensor, respectively, are configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r₁, r₂, r₃, r₄) as a part of the CSI report or they are known at the UE.

In accordance with a sub-embodiment 5 1 of the fifth embodiment 5, a UE is configured to report “explicit CSI Type V” with “delay-domain CSI for the higher-order singular-value matrix Ū_(S). In this configuration, the UE calculates an approximated reduced-sized (compressed) delay-domain higher-order singular-matrix Ũ_(S) from the frequency-domain higher-order singular-matrix Ū_(S). The delay-domain higher-order singular-matrix is given by Ū _(S) ≈F _(S) Ũ _(S), where F_(S)E

^(S×L) is an DFT matrix of size S×L. The size of the compressed delay-domain matrix Ũ_(S) is given by L×r₃. A compression is achieved when L<S.

The DFT vectors in F_(S) are selected from an oversampled DFT-codebook matrix Ω of dimension S×SO_(f). Here, O_(f)∈{1,2,3, . . . } denotes the oversampling factor of the DFT-codebook matrix. The indices of the selected vectors in F_(S) from the codebook Ω are stored in a set

=(i₁, i₂, . . . , i_(L)).

The UE quantizes the coefficients of the vectors in Û_(R)=[u_(R,1), . . . , u_(R,r) ₁ ]Ū_(T)=[u_(T,1), . . . ,u_(T,r) ₂ ], Ũ_(S)=[ũ_(S,1), . . . , ũ_(S,r) ₃ ]∈

^(L×r) ³ and Ū_(D)=[u_(D,1), . . . , u_(D,r) ₄ ] and the HO singular values s_(ijkl) using a codebook approach, and reports them along with the L delays, represented by a set of indices

that correspond to the selected DFT vectors in the codebook Ω, to the gNB.

The gNB reconstructs the frequency-domain channel tensor

_(c) according to this embodiment where Ū_(S) is calculated as Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ]=F_(S)Ũ_(S).

An illustration of this approach is shown in FIG. 13 .

In one method, the number of delays L is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred value of L as a part of the CSI report or it is known at the UE

The oversampled factor O_(f) of the DFT codebook matrix is configured via higher layer or via DCI physical layer signaling from the gNB to the UE or it is known at the UE.

In accordance with a sub-embodiment 5 2 of the fifth embodiment 5, a UE is configured to report “explicit CSI Type V” with “Doppler-frequency domain CSI for the higher-order singular-value matrix Ū_(D). In this configuration, the UE calculates an approximated reduced-sized (compressed) Doppler-frequency domain higher-order singular-matrix Ũ_(D) from the time domain higher-order singular-matrix Ū_(D).

The Doppler-frequency domain higher-order singular-matrix is given by Ū _(D) ≈F _(D) Ũ _(D), where F_(D)∈

^(D×G) is an DFT matrix of size D×G. The size of the compressed doppler-domain matrix Ũ_(D) is given by G×r₄. A compression is achieved when G<D.

The DFT vectors in F_(D) are selected from an oversampled DFT-codebook matrix of dimension D×DO_(t), where O_(t)∈{1,2,3, . . . } denotes the oversampling factor of the DFT-codebook matrix. The indices of the selected vectors in F_(D) from the codebook Ω are stored in a set

=(i₁, i₂, . . . , i_(G)).

The UE quantizes the coefficients of the vectors in Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ], Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ], Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ] and Ũ_(D)=[ũ_(D,1), . . . , ũ_(D,r) ₄ ]∈

^(G×r) ⁴ and the HO singular values s_(ijkl) using a codebook approach, and reports them along with the G Doppler-frequency values, represented by a set of indices

that correspond to the selected DFT vectors in the codebook Ω, to the gNB.

The gNB reconstructs the frequency-domain channel tensor

_(c) according to this embodiment, where Ū_(D) is calculated as Ū_(D)=[u_(D,1), . . . , u_(D,r) ₄ ]=F_(D)Ũ_(D).

An illustration of this approach is shown in FIG. 13 .

In one method, the number of Doppler-frequency values G is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred value of G as a part of the CSI report or it is known at the UE.

The oversampled factor O_(t) of the DFT codebook matrix is configured via higher layer or via physical layer signaling (via DCI) from the gNB to the UE or it is known at the UE.

Extension to Doppler Frequency Domain: Compression of Four-Dimensional Channel Tensor in Combination with HO-PCA

In accordance with a sixth embodiment 6, a UE is configured to report “explicit CSI Type VI” that represents a transformed and compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed covariance tensor over the configured subbands (SB), PRBs or subcarriers with respect to the space, frequency, time or frequency and space, or frequency and time, or space and time of the channel tensor. The CSI combines channel tensor transformation with data compression by exploiting the correlations of the channel coefficients in the spatial, frequency, delay and time/channel snapshot domain.

An illustration of this approach is shown in FIG. 14 . The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 4D frequency-domain channel tensor

∈

^(N) ^(r) ^(×N) ^(t) ^(×S×D) of dimension N_(r)×N_(t)×S×D, where S is the number of subbands, PRBs or subcarriers and D is number of snapshots of the channel measured at D consecutive time instants. The definition of N_(t) and N_(r) is dependent on the configuration of the CSI type:

-   -   N_(t) is the number of transmit antenna ports 2N₁N₂ for CSI type         configuration “channel tensor”, N_(t)=2N₁N₂, and N_(r) is the         number of UE receive antenna ports; N_(t) is the number of         transmit antenna ports 2N₁N₂ at the gNB, N_(t)=2N₁N₂,         N_(r)=2N₁N₂ for CSI type configuration “channel covariance         tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of UE receive antenna ports for CSI type         configuration “beamformed-channel tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of beamformed antenna ports/beams         N_(r)=2U for CSI type configuration “beamformed-channel         covariance tensor”;

After the construction of the frequency-domain channel tensor

, a one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D) or four-dimensional (4D) transformation of the channel tensor is applied with respect to the space, frequency, or time dimensions of the channel tensor. The aim of the transformation is to obtain a sparse or nearly-sparse representation of the channel tensor in one, two, three or four dimensions. After the transformation and compression step, the size of the channel tensor is reduced and a compression with respect to one, two or three dimensions or four dimensions of the channel tensor is achieved.

For example, a transformation/compression with respect to all four dimensions of the channel tensor

is represented by a (column-wise) Kronecker product as

${{\mspace{79mu}{{\mathcal{H} = {\mathcal{F}_{({1,2,3,4})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\quad\quad}}}\quad}{{vec}\left( {\mathcal{F}_{({1,2,3,4})}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r},n_{t},s,d} \otimes b_{1,n_{r},n_{t},s,d}}}}}}}},}}$ where

-   -   b_(1,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size N_(r)×1 with respect to the first dimension of the channel         tensor         , selected from a codebook matrix Ω₁;     -   b_(2,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size N_(t)×1 with respect to the second dimension of the channel         tensor         , selected from a codebook matrix Ω₂;     -   b_(3,n) _(r) _(,n) _(t) _(,s,d) transformation is a vector of         size S×1 with respect to the third dimension of the channel         tensor         , selected from a codebook matrix Ω₃;     -   b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size D×1 with respect to the fourth dimension of the channel         tensor         , selected from a codebook matrix Ω₄;     -   γ_(n) _(r) _(,n) _(t) _(,s,d) is the transformed/compressed         channel coefficient associated with the vectors b_(1,n) _(r)         _(,n) _(t) _(,s,d), b_(2,n) _(r) _(,n) _(t) _(,s,d)b_(3,n) _(r)         _(,n) _(t) _(,s,d), and b_(4,n) _(r) _(,n) _(t) _(,s,d) and     -   N′_(r), N′_(t), S′ and D′ represents the value of the first,         second, third and fourth dimension of the transformed/compressed         channel tensor         , respectively.

The transformed/compressed channel coefficients γ_(n) _(r) _(,n) _(t) _(,s,d) are used to form the transformed/compressed channel tensor

of dimension N′_(r)×N′_(t)×S′×D′, where N′_(r)≤N_(r), N′_(t)≤N_(t), S′≤S, D′≤D.

For example, a transformation/compression with respect to the two space dimensions of the channel tensor

is represented by

${{\mspace{79mu}{{\mathcal{H} = {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\quad\quad}}}\quad}{{vec}\left( {\mathcal{F}_{({1,2})}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,d} \otimes b_{3,s} \otimes b_{2,n_{r},n_{t},s,d} \otimes b_{1,n_{r},n_{t},s,d}}}}}}}},}}$ where b_(4,d) is a vector of all zeros with the d-th element being one, b_(3,s) is a vector of all zeros with the s-th element being one, b_(2,n) _(r) _(,n) _(t) _(,s,d) is a vector of size N_(t)×1 with respect to the second dimension of the channel tensor

, selected from a codebook matrix Ω₂, b_(1,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size N_(r)×1 with respect to the first dimension of the channel tensor

, selected from a codebook matrix Ω₁ and N′_(r)≤N_(r), N′_(t)≤N_(t), S′=S and D′=D.

For example, a transformation/compression with respect to the frequency and time dimension of the channel tensor

is represented by

${{\mspace{79mu}{{\mathcal{H} = {\mathcal{F}_{({3,4})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\quad\quad}}}\quad}{{vec}\left( {\mathcal{F}_{({3,4})}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}}$

Where b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size D×1 with respect to the fourth dimension of the channel tensor

, selected from a codebook matrix Ω₄, b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size S×1 with respect to the third dimension of the channel tensor

, selected from a codebook matrix Ω₃, b_(2,n) _(r) is a vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one and N′_(r)=N_(r) N′_(t)=N_(t), S′≤S and D′≤D.

For example, a transformation/compression with respect to the frequency dimension of the channel tensor

is represented by

${{\mspace{79mu}{{\mathcal{H} = {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\quad\quad}}}\quad}{{vec}\left( {\mathcal{F}_{(3)}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}}$ where b_(4,d) is a vector of all zeros with the d-th element being one, b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size S×1 with respect to the third dimension of the channel tensor

, selected from a codebook matrix Ω₃, b_(2,n) _(r) is a vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one and N′_(r)=N_(r) N′_(t)=N_(t), S′≤S and D′=D.

For example, a transformation/compression with respect to the time dimension of the channel tensor

is represented by

=

₍₄₎(

),

${{{{{vec}(\mathcal{H})} = {\quad\quad}}\quad}{{vec}\left( {\mathcal{F}_{(4)}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,s} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}}$ where b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size D×1 with respect to the fourth dimension of the channel tensor

, selected from a codebook matrix Ω₄, b_(3,s) is a vector of all zeros with the s-th element being one, b_(2,n) _(r) , is a vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one and N′_(r)=N_(r) N′_(t)=N_(t), S′=S and D′≤D.

As an example, the selection of the transformation/compression vectors and transformed/compressed channel coefficients for a transformation/compression with respect to the second and third dimension, can be calculated by min∥vec(vec(

)−vec(

)_((2,3))(

))∥₂ ².

The optimization problem may be solved by standard algorithms such as orthogonal matching pursuit. As a result, the indices of the vectors in the transformation matrices selected from the codebooks and the transformed channel coefficients associated with each domain are known.

The indices of the selected vectors b_(1,n) _(r) _(,n) _(t) _(,s,d), b_(2,n) _(r) _(,n) _(t) _(,s,d), b_(3,n) _(r) _(,n) _(t) _(,s,d) and b_(4,n) _(r) _(,n) _(t) _(,s,d) from the codebook matrices Ω_(n), n=1,2,3,4 are stored in a set

of g-tuples, where g refers to the number of transformed dimensions.

For example, for g=1, the set

is represented by

={i₁, i₂, . . . ,i_(T)}, where T denotes the number of selected vectors with respect to the transformed/compressed dimension of the channel tensor

. For example, in the case of a transformation/compression with respect to the frequency dimension, T=S′.

For example, for g=4, the set

is represented by 4-tuples (i_(1,n) _(r) , i_(2,n) _(t) , i_(3,s), i_(4,d)), where i_(1,n) _(r) is the index associated with vectors b_(1,n) _(r) _(,n) _(t) _(,s,d), i_(2,n) _(t) is the index associated with vector b_(2,n) _(r) _(,n) _(t) _(,s,d), i_(3,s) is the index associated with vector b_(3,n) _(r) _(,n) _(t) _(,s,d) and i_(4,d) is the index associated with vector b_(4,n) _(r) _(,n) _(t) _(,s,d). The set

is given by

={(i _(1,0) ,i _(2,0) ,i _(3,0) ,i _(4,0)), . . . ,(i _(1,N′) _(r) ,i _(2,N′) _(t) ,i _(3,S) ,i _(4,D′))}

In one method, the size of the set

is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred size of the set as a part of the CSI report or it is known at the UE.

The codebook matrices Ω_(n), are given by matrices Ω_(n)=[d_(n,0), d_(n,1), . . . , d_(n,TO) _(n) ⁻¹], where the parameter O_(f,n) denotes the oversampling factor with respect to the n-th dimension where T=N_(r) for n=1, T=N_(t) for n=2, T=S for n=3 and T=D′ for n=4.

The oversampled factors O_(f,n) of the codebook matrices are configured via higher layer (e.g., RRC, or MAC) or via DCI physical layer signaling from the gNB to the UE, or they are known at the UE or it is known at the UE.

After the channel transformation/compression, the UE performs a HO-PCA on the transformed/compressed channel tensor

, such that

is represented by

${\hat{\mathcal{H}} = {\sum\limits_{i = 1}^{N_{r}^{\prime}}\;{\sum\limits_{j = 1}^{N_{t}^{\prime}}{\sum\limits_{k = 1}^{S^{\prime}}{\sum\limits_{l = 1}^{D^{\prime}}{S_{ijkl}\left( {u_{R,i} \cdot u_{T,j} \cdot u_{S,k} \cdot u_{D,l}} \right)}}}}}},$ where

-   -   U_(R)=[u_(R,1), . . . , u_(R,N′) _(r) ]∈         ^(N′) ^(r) ^(×N′) ^(r) is a matrix containing the high-order         singular vectors with respect to the first dimension of the         transformed/compressed channel tensor         ;     -   U_(T)=[u_(T,1), . . . , u_(T,N′) _(t) ]∈         ^(N′) ^(t) ^(×N′) ^(t) is matrix containing the high-order         singular vectors with respect to the second dimension of the         transformed/compressed channel tensor         ;     -   U_(S)=[u_(S,1), . . . , u_(S,S′)]∈         ^(S′×S′) is a matrix containing the high-order singular vectors         with respect to the third dimension of the channel tensor         .     -   U_(D)=[u_(D,1), . . . , u_(D,R)]∈         ^(D′×R) is a matrix containing the higher order singular vectors         with respect to the time/channel snapshot dimension (fourth         dimension of the channel tensor         , where R is the rank of the channel tensor given by         R=min{N′_(r)N′_(t)S′, D′};     -   s_(ijkl) are the higher order singular values, sorted as         s_(ijkl)≥s_(i′j′k′l′) with i′≤j′≤j, k′≤k, l′≤l.

To reduce the number of channel coefficients, the channel tensor

is approximated by (r₁, r₂, r₃, r₄), (1≤r₁≤N′_(r), 1≤N′_(t), 1≤r₂≤N′_(t), 1≤r₃≤S′, 1≤r₄≤D′) dominant principal components with respect to the first, second, third and fourth dimensions and the corresponding 1-mode (left), 2-mode (right) and 3-mode (lateral) and 4-mode singular matrices. The compressed explicit frequency-domain channel tensor (explicit CSI) is given by

_(c)=Σ_(i=1) ^(r) ¹ Σ_(j=1) ^(r) ² Σ_(k=1) ^(r) ³ Σ_(l=1) ^(r) ⁴ s _(ijkl)(u _(R,i) ºu _(T,j) ºu _(S,k) ºu _(D,l)), where Ū _(R)=[u _(R,1) , . . . ,u _(R,r) ₁ ]∈

^(N′) ^(r) ^(×r) ¹ , ŪT=[u _(T,1) , . . . ,u _(T,r) ₂ ]∈

^(N′) ^(t) ^(×r) ² , Ū _(S)=[u _(S,1) , . . . ,u _(S,r) ₃ ]E

^(S′×r) ³ . Ū _(D)=[u _(D,1) , . . . ,u _(D,r) ₄ ]∈

^(D′×r) ⁴

The UE quantizes the coefficients of the vectors u_(R,i), u_(T,j), u_(S,k), u_(D,l) and the singular values s_(ijkl) using a codebook approach. The quantized vectors u_(R,i), u_(T,j), u_(S,k), u_(D,l) and the quantized singular values s_(ijkl) along with the index

set are reported to the gNB.

The gNB reconstructs first the transformed/compressed channel tensor

_(c) as

${\hat{\mathcal{H}}}_{c} = {\sum\limits_{i = 1}^{r_{1}}\;{\sum\limits_{j = 1}^{r_{2}}{\sum\limits_{k = 1}^{r_{3}}{\sum\limits_{l = 1}^{r_{4}}{{S_{ijkl}\left( {u_{R,i} \cdot u_{T,j} \cdot u_{S,k} \cdot u_{D,l}} \right)}.}}}}}$

Then, based on the transformed/compressed channel tensor

_(c) and the signaled index set

, the frequency-domain channel tensor

is reconstructed as vec(

)≈

_((1,2,3,4))(

_(c))(four-dimensional transformation/compression); vec(

)≈

_((1,2))(

_(c))(two-dimensional transformation/compression); vec(

)≈

_((3,4))(

_(c))(two-dimensional transformation/compression); vec(

)≈

₍₃₎(

_(c))(one-dimensional transformation/compression). vec(

)≈

₍₄₎(

_(c))(one-dimensional transformation/compression).

The number of complex coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by N′_(r)r₁+N′_(t)r₂+S′r₃+D′r₄+r₁r₂r₃r₄, for the higher-order singular vectors and the higher-order singular values.

In one method, the values of (r₁, r₂, r₃, r₄), representing the number of dominant principal components with respect to the first, second, third and fourth dimension of the transformed/compressed channel tensor

_(c), respectively, are configured via higher layer (e.g., RRC, or MAC-CE) signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r₁, r₂, r₃, r₄) as a part of the CSI report or they are known at the UE.

In accordance with a sub-embodiment 6 1 of the sixth embodiment 6, the 4D transformation/compression function

_((1,2,3,4))(

) is given by a two dimensional (2D-DCT) with respect to the space dimensions and a 2D-DFT transformation with respect to the frequency and time dimensions of the channel tensor. The codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices and the codebook matrix Ω₃ and Ω₄ are given by an oversampled DFT matrices.

In accordance with a sub-embodiment 6 2 of the sixth embodiment 6, the 4D transformation/compression function

_((1,2,3,4))(

) is given by a 4D-DFT transformation and the codebook matrices Ω_(n), n=1,2,3,4 are given by oversampled DFT matrices.

In accordance with a sub-embodiment 6 3 of the sixth embodiment 6, the 2D transformation/compression function

_((1,2))(

) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices.

In accordance with a sub-embodiment 6 4 of the sixth embodiment 6, the 2D transformation/compression function

_((3,4))(

) is given by a 2D-Discrete Fourier transformation (DFT) and the codebook matrices Ω_(n), n=3,4 are given by oversampled DFT matrices.

In accordance with a sub-embodiment 6 5 of the sixth embodiment 6, the 1D transformation/compression function

₍₃₎(

) is given by a 1D-DFT transformation and the codebook matrix Ω₃ is given by an oversampled DFT matrix.

In accordance with a sub-embodiment 6 6 of the sixth embodiment 6, the 1D transformation/compression function

₍₄₎(

) is given by a 1D-DFT transformation and the codebook matrix Ω₄ is given by an oversampled DFT matrix.

Extension to Doppler Frequency Domain: Compression of Four-Dimensional Channel Tensor in Combination with Non-HO-PCA (Standard PCA)

In accordance with a seventh embodiment 7, a UE is configured to report “explicit CSI Type VII” that represents a transformed and compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed covariance tensor over the configured subbands (SB), PRBs or subcarriers with respect to the space, frequency, time or frequency and space, or frequency and time, or space and time of the channel tensor. The CSI combines channel tensor transformation with data compression by exploiting the sparse representation in the delay domain and time domain and the correlations of the channel coefficients in the spatial and frequency/delay/time domains.

An illustration of this approach is shown in FIG. 15 . The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 4D frequency-domain channel

∈

^(N) ^(r) ^(×N) ^(t) ^(×S×D) of dimension N_(r)×N_(t)×S×D, where S is the number of subbands, PRBs or subcarriers and D is number of CSI-RS channel measurements over D consecutive time instants/slots. The definition of N_(t) and N_(r) is dependent on the configuration of the CSI type:

-   -   N_(t) is the number of transmit antenna ports 2N₁N₂ for CSI type         configuration “channel tensor”, N_(t)=2N₁N₂, and N_(r) is the         number of UE receive antenna ports; N_(t) is the number of         transmit antenna ports 2N₁N₂ at the gNB, N_(t)=2N₁N₂,         N_(r)=2N₁N₂, for CSI type configuration “channel covariance         tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of UE receive antenna ports for CSI type         configuration “beamformed-channel tensor”;     -   N_(t) is the number of beamformed antenna ports/beams N_(t)=2U         and N_(r) is the number of beamformed antenna ports/beams         N_(r)=2U for CSI type configuration “beamformed-channel         covariance tensor”;

After the construction of the frequency-domain channel tensor

, a one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D) or four-dimensional (4D) transformation of the channel tensor is applied with respect to the space, frequency, and/or time dimensions of the channel tensor. After the transformation and compression step, the size of the channel tensor is reduced and a compression with respect to one, two or three dimensions or four dimensions of the channel tensor is achieved.

For example, a transformation/compression with respect to all four dimensions of the channel tensor

is represented by a (column-wise) Kronecker product as

${{\mspace{79mu}{{\mathcal{H} = {\mathcal{F}_{({1,2,3,4})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\quad\quad}}}\quad}{{vec}\left( {\mathcal{F}_{({1,2,3,4})}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r},n_{t},s,d} \otimes b_{1,n_{r},n_{t},s,d}}}}}}}},}}$ where

-   -   b_(1,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size N_(r)×1 with respect to the first dimension of the channel         tensor         , selected from a codebook matrix Ω₁;     -   b_(2,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size N_(t)×1 with respect to the second dimension of the channel         tensor         , selected from a codebook matrix Ω₂;     -   b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size S×1 with respect to the third dimension of the channel         tensor         , selected from a codebook matrix Ω₃;     -   b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of         size D×1 with respect to the fourth dimension of the channel         tensor         , selected from a codebook matrix Ω₄;     -   γ_(n) _(r) _(,n) _(t) _(,s,d) is the transformed/compressed         channel coefficient associated with the vectors b_(1,n) _(r)         _(,n) _(t) _(,s,d), b_(2,n) _(r) _(,n) _(t) _(,s,d), b_(3,n)         _(r) _(,n) _(t) _(,s,d), and b_(4,n) _(r) _(,n) _(t) _(,s,d) and     -   N′_(r), N′_(t), S′ and D′ represents the value of the first,         second, third and fourth dimension of the transformed/compressed         channel tensor         , respectively.

The transformed/compressed channel coefficients γ_(n) _(r) _(,n) _(t) _(,s,d) are used to form the transformed/compressed channel tensor

of dimension N′_(r)×N′_(t)×S′×D′, where N′_(r)≤N_(r), N′_(t)≤N_(t), S′≤S, D′≤D.

For example, a transformation/compression with respect to the two space dimensions of the channel tensor

is represented by

${{\mspace{79mu}{{\mathcal{H} = {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\quad\quad}}}\quad}{{vec}\left( {\mathcal{F}_{({1,2})}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,d} \otimes b_{3,s} \otimes b_{2,n_{r},n_{t},s,d} \otimes b_{1,n_{r},n_{t},s,d}}}}}}}},}}$ where b_(4,d) is a vector of all zeros with the d-th element being one, b_(3,s) is a vector of all zeros with the s-th element being one, b_(2,n) _(r) _(,n) _(t) _(,s,d) is a vector of size N_(t)×1 with respect to the second dimension of the channel tensor

, selected from a codebook matrix Ω₂, b_(1,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size N_(r)×1 with respect to the first dimension of the channel tensor

, selected from a codebook matrix Ω₁ and N′_(r)≤N_(r), N′_(t)≤N_(t), S′=S and D′=D

For example, a transformation/compression with respect to the frequency and time dimension of the channel tensor

is represented by

${{\mspace{79mu}{{\mathcal{H} = {\mathcal{F}_{({3,4})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\quad\quad}}}\quad}{{vec}\left( {\mathcal{F}_{({3,4})}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}}$ where b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size D×1 with respect to the fourth dimension of the channel tensor

, selected from a codebook matrix Ω₄, b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size S×1 with respect to the third dimension of the channel tensor

, selected from a codebook matrix Ω₃, b_(2,n) _(r) is a vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one and N′_(r)=N_(r) N′_(t)=N_(t), S′≤S and D′≤D.

For example, a transformation/compression with respect to the frequency dimension of the channel tensor

is represented by

${{\mspace{79mu}{{\mathcal{H} = {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\quad\quad}}}\quad}{{vec}\left( {\mathcal{F}_{(3)}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}}$ where b_(4,d) is a vector of all zeros with the d-th element being one, b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size S×1 with respect to the third dimension of the channel tensor

, selected from a codebook matrix Ω₃, b_(2,n) _(r) is a vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one and N′_(r)=N_(r) N′_(t)=N_(t), S′≤S and D′=D.

For example, a transformation/compression with respect to the time dimension of the channel tensor

is represented by

=

₍₄₎(

),

${{{{{vec}(\mathcal{H})} = {\quad\quad}}\quad}{{vec}\left( {\mathcal{F}_{(4)}\left( \hat{H} \right)} \right)}} = {\quad{{{\quad\quad}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}\;{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,s} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},}}$ where b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size D×1 with respect to the fourth dimension of the channel tensor

,selected from a codebook matrix Ω₄, b_(3,s) is a vector of all zeros with the s-th element being one, b_(2,n) _(r) , is a vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one and N′_(r)=N_(r) N′_(t)=N_(t), S′=S and D′≤D.

As an example, the selection of the transformation/compression vectors and transformed/compressed channel coefficients for a transformation/compression with respect to the third and fourth dimension, can be calculated by min∥vec(

)−vec(

_((3,4))(

))∥₂ ².

The optimization problem may be solved by standard algorithms such as orthogonal matching pursuit. As a result, the indices of the vectors in the transformation matrices selected from the codebooks and the transformed channel coefficients associated with each domain are known.

The indices of the selected vectors b_(1,n) _(r) _(,n) _(t) _(,s,d), b_(2,n) _(r) _(,n) _(t) _(,s,d), b_(3,n) _(r) _(,n) _(t) _(,s,d) and b_(4,n) _(r) _(,n) _(t) _(,s,d) from the codebook matrices Ω_(n), n=1,2,3,4 are stored in a set

of g-tuples, where g refers to the number of transformed dimensions.

For example, for g=1, the set

is represented by

={i₁, i₂, . . . , i_(T)}, where T denotes the number of selected vectors with respect to the transformed/compressed dimension of the channel tensor

. For example, in the case of a transformation/compression with respect to the frequency dimension, T=S′.

For example, for g=4, the set

is represented by 4-tuples (i_(1,n) _(r) , i_(2,n) _(t) , i_(3,s), i_(4,d)), where i_(1,n) _(r) is the index associated with vectors b_(1,n) _(r) _(,n) _(t) _(,s,d), i_(2,n) _(t) , is the index associated with vector b_(2,n) _(r) _(,n) _(t) _(,s,d), i_(3,s) is the index associated with vector b_(3,n) _(r) _(,n) _(t) _(,s,d) and i_(4,d) is the index associated with vector b_(4,n) _(r) _(,n) _(t) _(,s,d). The set

is given by

={(i _(1,0) ,i _(2,0) ,i _(3,0) ,i _(4,0)), . . . ,(i _(1,N′) _(r) ,i _(2,N′) _(t) ,i _(3,S′) ,i _(4,D′))}

In one method, the size of the set

is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred size of the set as a part of the CSI report or it is known at the UE.

The codebook matrices Ω_(n), are given by matrices Ω_(n)=[d_(n,0), d_(n,1), . . . , d_(n,TO) _(n) ⁻¹], where the parameter O_(f,n) denotes the oversampling factor with respect to the n-th dimension with T being T=N_(r) for n=1, T=N_(t) for n=2, T=S for n=3 and T=D′ for n=4.

The oversampled factors O_(f,n) of the codebook matrices are configured via higher layer (e.g., RRC, or MAC) or via DCI physical layer signaling from the gNB to the UE, or they are known at the UE.

After the channel transformation/compression, the UE rewrites the transformed/compressed channel tensor to a transformed/compressed channel matrix

of size N′_(t)N′_(r)×S′D′, (see FIG. 12 ) and applies a standard PCA decomposition, represented by

${\hat{H} = {{U\;\Sigma\; V^{H}} = {\sum\limits_{i = 1}^{R}\;{s_{i}u_{i}v_{i}^{H}}}}},$ where

-   -   U=[u₁,u₂, . . . , u_(R)] is the N′_(t)N′_(r)×R left-singular         matrix;     -   V=[v₁, v₂, . . . , v_(R)] is the S′D′×R right-singular matrix;     -   Σ is a R×R diagonal matrix with ordered singular values s_(i)         (s₁≥s₂≥ . . . ≥s_(R)) on its main diagonal, and R=min(S′D′,         N′_(t)N′_(r)).

The transformed/compressed channel matrix

_(c) is then constructed using r, 1≤r≤R dominant principal components as

_(c) =ŪΣ V ^(H), where Ū=[u₁, u₂, . . . , u_(r),] and Σ=diag(s₁, s₂, . . . , s_(r)), and V=[v₁, v₂, . . . , v_(r)].

The UE quantizes Ū, V and the singular values s₁, s₂, . . . , s_(r) using a codebook approach, and then reports them along with the index set

to the gNB.

The gNB reconstructs first the transformed/compressed channel matrix

_(c) as

_(c) =ŪΣ V ^(H).

Then, based on the transformed/compressed channel matrix, the gNB constructs the transformed/compressed channel tensor

. Using the signaled index sets

, the frequency-domain channel tensor

, is reconstructed as vec(

)≈

_((1,2,3,4))(

_(c))(four-dimensional transformation/compression); vec(

)≈

_((1,2))(

_(c))(two-dimensional transformation/compression); vec(

)≈

_((3,4))(

_(c))(two-dimensional transformation/compression); vec(

)≈

₍₃₎(

_(c))(one-dimensional transformation/compression). vec(

)≈

₍₄₎(

_(c))(one-dimensional transformation/compression).

In one method, the value of r representing the number of dominant principal components of the transformed/compressed channel matrix

_(c) is configured via higher layer (e.g., RRC, or MAC-CE) signaling from the gNB to the UE. In another method, the UE reports the preferred value of r as part of the CSI report or it is known at the UE.

In accordance with a sub-embodiment 7 1 of the seventh embodiment 7, the 4D transformation/compression function

_((1,2,3,4))(

) is given by a two dimensional (2D-DCT) with respect to the space dimensions and a 2D-DFT transformation with respect to the frequency and time dimension of the channel tensor. The codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices and the codebook matrix Ω₃ and Ω₄ are given by an oversampled DFT matrices.

In accordance with a sub-embodiment 7 2 of the seventh embodiment 7, the 4D transformation/compression function

_((1,2,3,4))(

) is given by a 4D-DFT transformation and the codebook matrices Ω_(n), n=1,2,3,4 are given by oversampled DFT matrices.

In accordance with a sub-embodiment 7 3 of the seventh embodiment 7, the 2D transformation/compression function

_((1,2))(

) is given by a 2D-DCT and the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices.

In accordance with a sub-embodiment 7 4 of the seventh embodiment 7, the 2D transformation/compression function

_((3,4))(

) is given by a 2D-DFT and the codebook matrices Ω_(n), n=3,4 are given by oversampled DFT matrices.

In accordance with a sub-embodiment 7 5 of the seventh embodiment 7, the 1D transformation/compression function

₍₃₎(

) is given by a 1D-DFT transformation and the codebook matrix Ω₃ is given by an oversampled DFT matrix.

In accordance with a sub-embodiment 7 6 of the seventh embodiment 7, the 1D transformation/compression function

₍₄₎(

) is given by a 1D-DFT transformation and the codebook matrix Ω₄ is given by an oversampled DFT matrix.

CSI Type Configurations and Needed Signaling

In accordance with an eighth embodiment 8, the explicit CSI reporting is performed according to one of different proposed standard PCA or HO-PCA based CSI transformation/compression schemes. The gNB sends the explicit CSI report configuration to the UE. The explicit CSI report configuration contains

“explicit CSI Type I”:

-   -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the values (r₁, r₂, r₃) of dominant principal components with         respect to the first, second and third dimension of the channel         tensor;         “explicit CSI Type I” with “delay-domain CSI for the         higher-order singular matrix Ū_(S)”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the values (r₁, r₂, r₃) of dominant principal components with         respect to the first, second and third dimension of the channel         tensor;     -   the number of delays L reported by the UE;     -   the oversampling factor O_(f) of the DFT-codebook;         “explicit CSI Type II”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the value r of the dominant principal components of the channel         matrix;     -   the number of delays L reported by the UE;     -   the oversampling factor O_(f) of the DFT-codebook;         “explicit CSI Type III”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the transformation function type of the channel tensor         (“3D-DFT”, “2D-DCT”, “2D-DCT+1D-DFT”, “1D-DFT”);     -   the oversampling factors O_(f,n) of the codebooks with respect         to the three dimensions of the channel tensor;     -   the values (r₁, r₂, r₃) of dominant principal components with         respect to the first, second and third dimension of the         transformed/compressed channel tensor;         “explicit CSI Type IV”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the transformation function type of the channel tensor         (“3D-DFT”, “2D-DCT”, “2D-DCT+1D-DFT”, “1D-DFT”)     -   the oversampling factors O_(f,n) of the codebooks with respect         to the three dimensions of the channel tensor;     -   the value r of the dominant principal components of the         transformed/compressed channel matrix;         “explicit CSI Type V”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the values (r₁, r₂, r₃, r₄) of dominant principal components         with respect to the first, second, third and fourth dimension of         the channel tensor;         “explicit CSI Type V” with “delay-domain CSI for the         higher-order singular matrix IL”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the values (r₁, r₂, r₃, r₄) of dominant principal components         with respect to the first, second, third and fourth dimension of         the channel tensor;     -   the number of delays L reported by the UE;     -   the oversampling factor O_(f) of the DFT-codebook;         “explicit CSI Type V” with “time/Doppler-frequency-domain CSI         for the higher-order singular matrix Ū_(D)”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the values (r₁, r₂, r₃, r₄) of dominant principal components         with respect to the first, second, third and fourth dimension of         the channel tensor;     -   the number of doppler frequencies G reported by the UE;     -   the oversampling factor O_(t) of the DFT-codebook;         “explicit CSI Type VI”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the transformation function type of the channel tensor         (“4D-DFT”, “2D-DCT”, “2D-DCT+2D-DFT”, “2D-DFT for frequency- and         time/Doppler-frequency domain”, “1D-DFT for         time/Doppler-frequency-domain”, “1D-DFT for frequency-domain”)     -   the oversampling factors O_(f,n) of the codebooks with respect         to the four dimensions of the channel tensor;     -   the values (r₁, r₂, r₃, r₄) of dominant principal components         with respect to the first, second, third and fourth dimension of         the transformed/compressed channel tensor;         “explicit CSI Type VII”:     -   CSI channel type (“channel”, “covariance of channel”,         “beamformed-channel”, “covariance of beamformed-channel”);     -   the transformation function type of the channel tensor         (“4D-DFT”, “2D-DCT”, “2D-DCT+2D-DFT”, “2D-DFT for frequency- and         time/Doppler-frequency domain”, “1D-DFT for         time/Doppler-frequency domain”, “1D-DFT for frequency-domain”)     -   the oversampling factors O_(f,n) for the codebooks with respect         to the four dimensions of the channel tensor;     -   the value r of the dominant principal components of the         transformed/compressed channel matrix;

In response, the UE

-   -   performs measurements of CSI-RS over D time instants/slots [if D         is configured]     -   constructs the channel tensor or channel matrix depending on the         configured CSI channel type for each SB in which it is         configured to report explicit CSI;     -   applies a transformation/compression function to the channel         tensor or channel matrix and calculates the index set         depending on the configuration as explained with reference to         embodiments 1-7;     -   performs a standard PCA on the channel matrix, or an HO-PCA on         the channel tensor depending on the configuration as explained         with reference to embodiments 1-7;     -   finally quantizes the singular matrices, singular values and         reports them along with the index set         to the gNB.

The gNB reconstructs the channel tensor or channel matrix as explained with reference to embodiments 1-7.

Codebook for Singular-Value Matrices and Singular Values Quantization

In accordance with a ninth embodiment 9, a UE is configured with separate codebooks, or a joint codebook for the quantization of

“explicit CSI Type I”

-   -   entries of each vector of Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ];     -   entries of each vector of Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ];     -   entries of each vector of Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ];     -   singular values s_(ijk)         “explicit CSI Type I” in combination with “delay-domain CSI for         the higher-order singular matrix U”:     -   entries of each vector of Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ];     -   entries of each vector of Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ];     -   entries of each vector of Ū_(S)=[ũ_(S,1), . . . , ũ_(S,r) ₃ ];     -   singular values s_(ijk)         “explicit CSI Type II”:     -   entries of each vector of Ū=[u₁, u₂, . . . , u_(r)];     -   entries of each vector of {tilde over (V)}=[{tilde over (v)}₁,         {tilde over (v)}₂, . . . , {tilde over (v)}_(r)];     -   singular values Σ=diag(s₁, s₂, . . . , s_(r))         “explicit CSI Type III”:     -   entries of each vector of Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ];     -   entries of each vector of Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ];     -   entries of each vector of Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ];     -   singular values s_(ijk)         “explicit CSI Type IV”:     -   entries of each vector of Ū=[u₁, u₂, . . . , u_(r)];     -   entries of each vector of V=[v₁, v₂, . . . , v_(r)];     -   singular values Σ=diag(s₁, s₂, . . . , s_(r))         “explicit CSI Type V”:     -   entries of each vector of Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ];     -   entries of each vector of Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ];     -   entries of each vector of Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ];     -   entries of each vector of Ū_(D)=[u_(D,1), . . . , u_(D,r) ₄ ];     -   singular values s_(ijkl)         “explicit CSI Type V” in combination with “delay-domain CSI for         the higher-order singular matrix Ū_(S)”:     -   entries of each vector of Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ];     -   entries of each vector of Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ];     -   entries of each vector of Ū_(S)=[ũ_(S,1), . . . , ũ_(S,r) ₃ ];     -   entries of each vector of Ū_(D)=[u_(D,1), . . . , u_(D,r) ₄ ];     -   singular values s_(ijkl)         “explicit CSI Type V” in combination with “Doppler-frequency         domain CSI for the higher-order singular matrix Ū_(D)”:     -   entries of each vector of Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ];     -   entries of each vector of Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ];     -   entries of each vector of Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ];     -   entries of each vector of Ū_(D)=[ũ_(D,1), . . . , ũ_(D,r) ₄ ];     -   singular values s_(ijkl)         “explicit CSI Type “VI”:     -   entries of each vector of Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ];     -   entries of each vector of Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ];     -   entries of each vector of Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ]     -   entries of each vector of Ū_(D)=[u_(D,1), . . . , u_(D,r) ₄ ];     -   singular values s_(ijkl)         “explicit CSI Type “VII”:     -   entries of each vector of Ū=[u₁, u₂, . . . , u_(r)]     -   entries of each vector of V=[v₁, v₂, . . . v_(r)];     -   singular values Σ=diag(s₁, s₂, . . . , s_(r))         according to the following embodiments.

In accordance with a sub-embodiment 9 1 of the ninth embodiment 9, a UE is configured with a scalar codebook for the quantization of each entry of vector of the HO-PCA, or standard-PCA singular-matrices according to the following alternatives:

-   -   Common codebook: each entry of each HO-PCA, or standard-PCA         singular-matrix is quantized with the same resolution/codebook,         with k bits for the amplitude and n bits for the phase;     -   Separate codebooks: the entries of each HO-PCA, or standard-PCA         singular-matrix are quantized with different         resolutions/codebooks. For example, for the HO-PCA channel         compression, entries of each vector of Ū_(R) are quantized with         k₁ bits for the amplitude and n₁ bits for the phase, and entries         of each vector of Ū_(T) are quantized with k₂ bits for the         amplitude and n₂ bits for the phase, and entries of each vector         of Ū_(S) are quantized with k₃ bits for the amplitude and n₃         bits for the phase, and entries of each vector of Ū_(D) are         quantized with k₄ bits for the amplitude and n₄ bits for the         phase.

The codebook(s) is/are a priori known by the UE, or configured via higher layer signaling from the gNB to the UE.

In accordance with a sub-embodiment 9 2 of the ninth embodiment 9, a UE is configured with unit-norm vector codebook(s) for the quantization of each vector of the HO-PCA, or standard-PCA singular-matrices.

Let C_(Ū) _(R) , C_(Ū) _(T) , C_(Ū) _(S) , C_(Ũ) _(D) , C_({tilde over (S)}) _(D) and C_(Ū), C _(V) , C_({tilde over (V)}) be codebooks for high order singular matrices Ū_(R), Ū_(T), Ū_(S) Ū_(D), Ũ_(S) and the singular matrices Ū, V, {tilde over (V)}, respectively, where depending on the explicit CSI configuration,

“explicit CSI Type I”:

-   -   Codebook C_(Ū) _(R) comprises a set of unit-norm vectors, each         of size N_(r)×1,     -   Codebook C_(Ū) _(T) comprises a set of unit-norm vectors, each         of size N_(t)×1, and     -   Codebook C_(Ū) _(S) comprises a set of unit-norm vectors, each         of size S×1.         “explicit CSI Type I” in combination with “delay-domain CSI for         the higher-order singular matrix U”:     -   Codebook C_(Ū) _(R) comprises a set of unit-norm vectors, each         of size N_(r)×1,     -   Codebook C_(Ū) _(T) comprises a set of unit-norm vectors, each         of size N_(t)×1, and     -   Codebook C_(Ũ) _(S) comprises a set of unit-norm vectors, each         of size L×1.         “explicit CSI Type II”:     -   Codebook C_(Ū) comprises a set of unit-norm vectors, each of         size N_(t)N_(r)×1,     -   Codebook C_({tilde over (V)}) comprises a set of unit-norm         vectors, each of size L×1.         “explicit CSI Type III”:     -   Codebook C_(Ū) _(R) comprises a set of unit-norm vectors, each         of size N′_(r)×1,     -   Codebook C_(Ū) _(T) comprises a set of unit-norm vectors, each         of size N′_(t)×1, and     -   Codebook C_(Ū) _(S) comprises a set of unit-norm vectors, each         of size S′×1.         “explicit CSI Type IV”:     -   Codebook C_(Ū) comprises a set of unit-norm vectors, each of         size N′×1,     -   Codebook C _(V) comprises a set of unit-norm vectors, each of         size S′×1.         “explicit CSI Type V”:     -   Codebook C_(Ū) _(R) comprises a set of unit-norm vectors, each         of size N_(r)×1,     -   Codebook C_(Ū) _(T) comprises a set of unit-norm vectors, each         of size N_(t)×1, and     -   Codebook C_(Ū) _(S) comprises a set of unit-norm vectors, each         of size S×1.     -   Codebook C_(Ū) _(D) comprises a set of unit-norm vectors, each         of size D×1.         “explicit CSI Type V” in combination with “delay-domain CSI for         the higher-order singular matrix Ū_(S)”:     -   Codebook C_(Ū) _(R) comprises a set of unit-norm vectors, each         of size N_(r)×1,     -   Codebook C_(Ū) _(T) comprises a set of unit-norm vectors, each         of size N_(t)×1,     -   Codebook C_(Ũ) _(S) comprises a set of unit-norm vectors, each         of size L×1.     -   Codebook C_(Ū) _(D) comprises a set of unit-norm vectors, each         of size D×1.         “explicit CSI Type V” in combination with “d         time/Doppler-frequency-domain CSI for the higher-order singular         matrix Ū_(n)”:     -   Codebook C_(Ū) _(R) comprises a set of unit-norm vectors, each         of size N_(r)×1,     -   Codebook C_(Ū) _(T) comprises a set of unit-norm vectors, each         of size N_(t)×1,     -   Codebook C_(Ū) _(S) comprises a set of unit-norm vectors, each         of size S×1.     -   Codebook C_(Ũ) _(D) comprises a set of unit-norm vectors, each         of size G×1.         “explicit CSI Type VI”:     -   Codebook C_(Ū) _(R) comprises a set of unit-norm vectors, each         of size N′_(r)×1,     -   Codebook C_(Ū) _(T) comprises a set of unit-norm vectors, each         of size N′_(t)×1, and     -   Codebook C_(Ū) _(S) comprises a set of unit-norm vectors, each         of size S′×1.     -   Codebook C_(Ū) _(D) comprises a set of unit-norm vectors, each         of size D′×1.         “explicit CSI Type VII”:     -   Codebook C_(Ū) comprises a set of unit-norm vectors, each of         size N′_(t)N′×1,     -   Codebook C _(V) comprises a set of unit-norm vectors, each of         size S′D′×1.

The UE selects for each vector/column in matrix A (where A represents one of the following matrices Ū_(R), Ū_(T), Ū_(S), Ũ_(S), Ū_(D), Ũ_(D), or Ū, V, {tilde over (V)}) a vector in C_(A) to represent the vector/column in matrix A, and reports the indices corresponding to the selected vectors in C_(A) as a part of the CSI report to the gNB.

In accordance with a sub-embodiment 9 3 of the ninth embodiment 9, a UE is configured with a scalar codebook to quantize the high-order singular values s_(ijk) or s_(ijk) the non-HO singular values in matrix Σ using a scalar codebook, where each singular value is quantized with k bits for the amplitude.

Overhead Reduction

In accordance with a tenth embodiment 10, a UE is configured with explicit CSI reporting as described in embodiments 1-7 and to reduce overhead the UE reports, depending on the explicit CSI configuration, the HO singular matrices (Ū_(R), Ū_(T), Ū_(S) or Ũ_(S), Ū_(D) or Ũ_(D)) or the non-HO singular matrices (Ū, V, or {tilde over (V)}), separately in alternative CSI reporting instances.

CSI-RS-BurstDuration

In accordance with further embodiments, the gNB or base station sends a CSI-RS configuration and CSI report configuration to the UE, and the CSI-RS configuration may include a CSI-RS resource(s) configuration with respect to sub-clause 7.4.1.5 in TS 38.211 (3GPP TS 38.211 V15.1.0, “3rd Generation Partnership Project; Technical Specification Group Radio Access Network; NR; Physical channels and modulation (Release 15), March 2018) and with sub-clause 6.3.2 in TS.38.331 (3GPP TS 38.331 V15.1.0, “3rd Generation Partnership Project; Technical Specification Group Radio Access Network; NR; Radio Resource Control (RRC); Protocol specification (Release 15), March 2018. Further, an additional higher layer parameter configuration referred to as CSI-RS-BurstDuration is included.

The CSI-RS-BurstDuration is included to provide a CSI-RS design allowing to track the time-evolution of the channel. In accordance with embodiments, a UE is configured with a CSI-RS resource set(s) configuration with the higher layer parameter CSI-RS-BurstDuration, in addition to the configurations from clause 7.4.1.5 in TS 38.211 and clause 6.3.2 in TS.38.331 mentioned above, to track the time-evolution of CSI. The time-domain-repetition of the CSI-RS, in terms of the number of consecutive slots the CSI-RS is repeated in, is provided by the higher layer parameter CSI-RS-BurstDuration. The possible values of CSI-RS-BurstDuration for the NR numerology μ are 2^(μ)·X_(B) slots, where X_(B)∈{0,1,2, . . . , maxNumBurstSlots−1}. The NR numerology μ=0,1,2,3,4 . . . defines, e.g., a subcarrier spacing of 2^(μ)·15 kHz in accordance with the NR standard.

For example, when the value of X_(B)=0 or the parameter CSI-RS-BurstDuration is not configured, there is no repetition of the CSI-RS over multiple slots. The burst duration scales with the numerology to keep up with the decrease in the slot sizes. Using the same logic used for periodicity of CSI-RS. FIG. 16(a) illustrates a CSI-RS with a periodicity of 10 slots and no repetition (CSI-RS-BurstDuration not configured or CSI-RS-BurstDuration=0), and FIG. 16(b) illustrates a CSI-RS with a periodicity of 10 slots and repetition of 4 slots (CSI-RS-BurstDuration=4). FIG. 17 illustrates a CSI-RS-BurstDuration information element in accordance with an embodiment. The information element of the new RRC parameter CSI-RS-BurstDuration is as follows: the value next to the text burstSlots indicates the value of X_(B), which for a given New Radio numerology μ (see [1]) provides the burst duration 2^(μ)·X_(B) of the CSI-RS, i.e., the number of consecutive slots of CSI-RS repetition.

The burst-CSI-RS across multiple consecutive slots enables the extraction of time-evolution information of the CSI and for reporting the explicit CSI, in a way as described in more detail above. In other words, the UE may calculate the explicit CSI according to the embodiments described above with a repetition of the CSI-RS resource(s) over multiple consecutive slots, and report them accordingly.

In accordance with the embodiments, the wireless communication system may include a terrestrial network, or a non-terrestrial network, or networks or segments of networks using as a receiver an airborne vehicle or a spaceborne vehicle, or a combination thereof.

In accordance with the embodiments, the UE may comprise one or more of a mobile or stationary terminal, an IoT device, a ground based vehicle, an aerial vehicle, a drone, a building, or any other item or device provided with network connectivity enabling the item/device to communicate using the wireless communication system, like a sensor or actuator.

In accordance with the embodiments, the base station may comprise one or more of a macro cell base station, or a small cell base station, or a spaceborne vehicle, like a satellite or a space, or an airborne vehicle, like a unmanned aircraft system (UAS), e.g., a tethered UAS, a lighter than air UAS (LTA), a heavier than air UAS (HTA) and a high altitude UAS platforms (HAPs), or any transmission/reception point (TRP) enabling an item or a device provided with network connectivity to communicate using the wireless communication system.

In accordance with embodiments, the wireless communication system may include a terrestrial network, or a non-terrestrial network, or networks or segments of networks using as a receiver an airborne vehicle or a spaceborne vehicle, or a combination thereof.

In accordance with embodiments, the UE may comprise one or more of a mobile or stationary terminal, an IoT device, a ground based vehicle, an aerial vehicle, a drone, a building, or any other item or device provided with network connectivity enabling the item/device to communicate using the wireless communication system, like a sensor or actuator.

In accordance with embodiments, the base station may comprise one or more of a macro cell base station, or a small cell base station, or a spaceborne vehicle, like a satellite or a space, or an airborne vehicle, like a unmanned aircraft system (UAS), e.g., a tethered UAS, a lighter than air UAS (LTA), a heavier than air UAS (HTA) and a high altitude UAS platforms (HAPs), or any transmission/reception point (TRP) enabling an item or a device provided with network connectivity to communicate using the wireless communication system.

The embodiments of the present invention have been described above with reference to a communication system employing a rank 1 or layer 1 communication. However, the present invention is not limited to such embodiments and may also be implemented in a communication system employing a higher rank or layer communication. In such embodiments, the feedback includes the delays per layer and the complex precoder coefficients per layer.

The embodiments of the present invention have been described above with reference to a communication system in which the transmitter is a base station serving a user equipment, and the communication device or receiver is the user equipment served by the base station. However, the present invention is not limited to such embodiments and may also be implemented in a communication system in which the transmitter is a user equipment station, and the communication device or receiver is the base station serving the user equipment. In accordance with other embodiments, the communication device and the transmitter may both be UEs communicating via directly, e.g., via a sidelink interface.

Although some aspects of the described concept have been described in the context of an apparatus, it is clear that these aspects also represent a description of the corresponding method, where a block or a device corresponds to a method step or a feature of a method step. Analogously, aspects described in the context of a method step also represent a description of a corresponding block or item or feature of a corresponding apparatus.

Various elements and features of the present invention may be implemented in hardware using analog and/or digital circuits, in software, through the execution of instructions by one or more general purpose or special-purpose processors, or as a combination of hardware and software. For example, embodiments of the present invention may be implemented in the environment of a computer system or another processing system. FIG. 18 illustrates an example of a computer system 350. The units or modules as well as the steps of the methods performed by these units may execute on one or more computer systems 350. The computer system 350 includes one or more processors 352, like a special purpose or a general purpose digital signal processor. The processor 352 is connected to a communication infrastructure 354, like a bus or a network. The computer system 350 includes a main memory 356, e.g., a random access memory (RAM), and a secondary memory 358, e.g., a hard disk drive and/or a removable storage drive. The secondary memory 358 may allow computer programs or other instructions to be loaded into the computer system 350. The computer system 350 may further include a communications interface 360 to allow software and data to be transferred between computer system 350 and external devices. The communication may be in the from electronic, electromagnetic, optical, or other signals capable of being handled by a communications interface. The communication may use a wire or a cable, fiber optics, a phone line, a cellular phone link, an RF link and other communications channels 362.

The terms “computer program medium” and “computer readable medium” are used to generally refer to tangible storage media such as removable storage units or a hard disk installed in a hard disk drive. These computer program products are means for providing software to the computer system 350. The computer programs, also referred to as computer control logic, are stored in main memory 356 and/or secondary memory 358. Computer programs may also be received via the communications interface 360. The computer program, when executed, enables the computer system 350 to implement the present invention. In particular, the computer program, when executed, enables processor 352 to implement the processes of the present invention, such as any of the methods described herein. Accordingly, such a computer program may represent a controller of the computer system 350. Where the disclosure is implemented using software, the software may be stored in a computer program product and loaded into computer system 350 using a removable storage drive, an interface, like communications interface 360.

The implementation in hardware or in software may be performed using a digital storage medium, for example cloud storage, a floppy disk, a DVD, a Blue-Ray, a CD, a ROM, a PROM, an EPROM, an EEPROM or a FLASH memory, having electronically readable control signals stored thereon, which cooperate (or are capable of cooperating) with a programmable computer system such that the respective method is performed. Therefore, the digital storage medium may be computer readable.

Some embodiments according to the invention comprise a data carrier having electronically readable control signals, which are capable of cooperating with a programmable computer system, such that one of the methods described herein is performed.

Generally, embodiments of the present invention may be implemented as a computer program product with a program code, the program code being operative for performing one of the methods when the computer program product runs on a computer. The program code may for example be stored on a machine readable carrier.

Other embodiments comprise the computer program for performing one of the methods described herein, stored on a machine readable carrier. In other words, an embodiment of the inventive method is, therefore, a computer program having a program code for performing one of the methods described herein, when the computer program runs on a computer.

A further embodiment of the inventive methods is, therefore, a data carrier (or a digital storage medium, or a computer-readable medium) comprising, recorded thereon, the computer program for performing one of the methods described herein. A further embodiment of the inventive method is, therefore, a data stream or a sequence of signals representing the computer program for performing one of the methods described herein. The data stream or the sequence of signals may for example be configured to be transferred via a data communication connection, for example via the Internet. A further embodiment comprises a processing means, for example a computer, or a programmable logic device, configured to or adapted to perform one of the methods described herein. A further embodiment comprises a computer having installed thereon the computer program for performing one of the methods described herein.

In some embodiments, a programmable logic device (for example a field programmable gate array) may be used to perform some or all of the functionalities of the methods described herein. In some embodiments, a field programmable gate array may cooperate with a microprocessor in order to perform one of the methods described herein. Generally, the methods are performed by any hardware apparatus.

While this invention has been described in terms of several embodiments, there are alterations, permutations, and equivalents which fall within the scope of this invention. It should also be noted that there are many alternative ways of implementing the methods and compositions of the present invention. It is therefore intended that the following appended claims be interpreted as including all such alterations, permutations and equivalents as fall within the true spirit and scope of the present invention. 

The invention claimed is:
 1. A communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, and a processor configured to estimate the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, construct a frequency-domain channel tensor using the CSI estimate, perform a higher-order principal component analysis, HO-PCA, on the channel tensor, identify a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and report to the transmitter the explicit CSI comprising the dominant principal components of the channel tensor.
 2. The communication device of claim 1, wherein the communication device is configured to receive from the transmitter an explicit CSI report configuration comprising a CSI channel-type information, CSI-Ind, indicator for the CSI report, wherein the CSI-Ind indicator is associated with a channel-type configuration, the channel tensor is a three-dimensional, 3D, channel tensor, or is represented by a 3D channel covariance tensor, a 3D beamformed-channel tensor, or a 3D beamformed channel covariance tensor, as indicated by the CSI-Ind indicator, and wherein the plurality of dominant principal components of the compressed 3D channel tensor of dimension N_(r)×N_(t)×S comprise: a first set of r₁ basis vectors comprised by a matrix Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ]∈

^(N) ^(r) ^(×r) ¹ ; a second set of r₂ basis vectors comprised by a matrix Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ]∈

^(N) ^(t) ^(×r) ² ; a third set of r₃ basis vectors comprised by a matrix Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ]∈

^(S×r) ³ ; and r₁r₂r₃associated high-order singular values s_(ijk), sorted such that Σ_(j,k)|s_(i,j,k)|²≥Σ_(j,k)|s_(i+1,j,k)|², Σ_(i,k)|s_(i,j,k)|²≥Σ_(i,k)|s_(i,j+1,k)|², and Σ_(i,j)|s_(i,j,k)|²≥Σ_(i,j)|s_(i,j,k+1)|² for all i, j, k.
 3. The communication device of claim 2, wherein the values of r₁, r₂, and r₃ representing the number of dominant principal components with respect to the first, second and third dimension of the compressed 3D channel tensor, respectively, are configured via the CSI report configuration by the transmitter, or reported by the communication device in the CSI report, or pre-determined and known at the communication device.
 4. The communication device of claim 2, wherein the processor is configured to quantize the coefficients in the vectors u_(R,i), u_(T,j), and u_(S,k) and the HO singular values s_(ijk) of the 3D channel tensor using a codebook approach, the number of complex coefficients to be quantized being given by N_(r)r₁+N_(t)r₂+Sr₃ for the higher-order singular vectors, and a number of real coefficients to be quantized being given by r₁r₂r₃ for the higher-order singular values, respectively.
 5. The communication device of claim 1, wherein the channel tensor is a four-dimensional, 4D, channel tensor, or represented either by a 4D channel covariance tensor, a 4D beamformed-channel tensor, or a 4D beamformed channel covariance tensor, as indicated by the CSI-Ind indicator, and wherein the plurality of dominant principal components of the compressed 4D channel tensor of dimension N_(r)×N_(t)×S×D comprise: a first set of r₁ basis vectors comprised by a matrix Ū_(R)=[u_(R,1), . . . , u_(R,r) ₁ ]∈

^(N) ^(r) ^(×r) ¹ ; a second set of r₂ basis vectors comprised by a matrix Ū_(T)=[u_(T,1), . . . , u_(T,r) ₂ ]∈

^(N) ^(t) ^(×r) ² ; a third set of r₃ basis vectors comprised by a matrix Ū_(S)=[u_(S,1), . . . , u_(S,r) ₃ ]∈^(S×r) ³ ; a fourth set of r₄ basis vectors comprised by a matrix Ū_(D)=[u_(D,1), . . . ,u_(D,r) ₄ ]∈

^(D×r) ⁴ ; and r₁r₂r₃r₄ associated high-order singular values s_(ijkl), sorted such that Σ_(j,k,l)|s_(i,j,k,l)|²≥Σ_(j,k,l)|s_(i+1,j,k,l)|², Σ_(i,k,l)|s_(i,j,k,l)|²≥Σ_(i,k,l)|s_(i,j+1,k,l)|², Σ_(i,j,l)|s_(i,j,k,l)|²≥Σ_(i,j,l)|s_(i,j,k+1,l)|², and Σ_(i,j,k)|s_(i,j,k,l)|²≥Σ_(i,j,k)|s_(i,j,k,l+1)|² for all i, j, k, l.
 6. The communication device of claim 5, wherein the values of r₁,r₂, r₃, and r₄ representing the number of dominant principal components of the 4D channel tensor are configured via the CSI report configuration by the transmitter, or they are reported by the communication device in the CSI report, or they are pre-determined and known at the communication device.
 7. The communication device of claim 5, wherein the processor is configured to quantize the coefficients of the vectors u_(R,i), u_(T,j), u_(S,k), u_(D,l) and the singular values s_(ijkl) using a codebook approach, a number of complex coefficients to be quantized being given by N_(r)r₁+N_(t)r₂+Sr₃+Dr₄ for the higher-order singular vectors, and a number of real coefficients to be quantized being given by r₁r₂r₃r₄ for the higher-order singular values, respectively.
 8. The communication device of claim 1, wherein the explicit CSI comprises a delay-domain CSI for the higher-order singular-value matrix Ū_(S), and wherein the processor is configured to calculate a reduced-sized delay-domain higher-order singular-matrix Ũ_(S) from the frequency-domain higher-order singular-matrix Ū_(S), wherein the delay-domain higher-order singular-matrix is given by Ū _(S) ≈F _(S) Ũ _(S), where F_(S)∈

^(S×L) contains L vectors of size S×1, selected from a discrete Fourier transform, DFT, codebook Ω, the size of the compressed delay-domain matrix Ũ_(S) is given by L×r₃, and L is the number of delays, and the oversampled codebook matrix is given by Ω=[d₀, d₁, . . . , d_(SO) _(f) ⁻¹], where d_(i)= ${\left\lbrack {1e^{\frac{{- j}2\pi i}{o_{f}s}}\ldots e^{\frac{{- j}2\pi{i({S - 1})}}{o_{f}s}}} \right\rbrack^{T} \in {\mathbb{C}}^{S \times 1}},{i \in \left\{ {0,\ldots,{{SO}_{f} - 1}} \right\}},$ and O_(f)∈{1,2,3, . . . } denotes the oversampling factor of the DFT-codebook matrix; quantize the coefficients in vectors Ũ_(S)=[ũ_(S,1), . . . , ũ_(S,r) ₃ ]∈

^(L×r) ³ using a codebook approach; report to the transmitter the explicit CSI comprising the coefficients of Ũ_(S) instead of Ū_(S), along with the L delays, represented by a set of indices that correspond to the selected DFT-vectors in the codebook Ω.
 9. The communication device of claim 1, wherein the explicit CSI comprises a Doppler-frequency domain CSI for the higher-order singular-value matrices Ū_(D), and wherein the processor is configured to calculate a reduced-sized Doppler-frequency domain higher-order singular-matrix Ũ_(D) from the time-domain higher-order singular-matrix Ū_(D), wherein the Doppler-frequency domain higher-order singular-matrix is given by Ū _(D) ≈F _(D) Ũ _(D), where F_(D)∈

^(D×G) contains G vectors of size D×1, selected from a discrete Fourier transform, DFT, codebook Ω, the size of the compressed Doppler-frequency domain matrix Ũ_(D) is given by G×r₄, and G is the number of Doppler-frequency components, and the oversampled codebook matrix is given by Ω=[d₀, d₁, . . . , d_(DO) _(t) ⁻¹], where ${d_{i} = {{\left\lbrack {1e^{\frac{{- j}2\pi i}{o_{t}D}}\ldots e^{\frac{{- j}2\pi{i({D - 1})}}{o_{t}D}}} \right\rbrack^{T} \in}{\mathbb{C}}^{D \times 1}}},{i \in \left\{ {0,\ldots,{{O_{t}D} - 1}} \right\}},$ and O_(t)∈{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; quantize the coefficients in vectors Ũ_(D)[ũ_(D,1), . . . , ũ_(D,r) ₄ ]∈

^(G×r) ⁴ using a codebook approach; or report to the transmitter the explicit CSI report comprising the coefficients of Ũ_(D) instead of Ū_(D), along with the G Doppler-frequency components, represented by a set of indices that correspond to the selected DFT-vectors in the codebook Ω.
 10. The communication device of claim 1, wherein the processor is configured to apply, after the construction of a 3D channel tensor, a one-dimensional, a two-dimensional, or multi-dimensional transformation/compression of the channel tensor with respect to the space dimension of the 3D channel tensor, or the frequency dimension of the 3D channel tensor, or both the frequency and space dimensions of the 3D channel tensor, or after the construction of a 4D channel tensor, a one-dimensional, a two-dimensional, or multi-dimensional transformation/compression of the channel tensor with respect to the space dimension of the channel tensor, or the frequency dimension of the channel tensor, or the time dimension of the channel tensor, or both the frequency and time dimensions of the channel tensor, so as to exploit a sparse or nearly-sparse representation of the 3D or 4D channel tensor in one or more dimensions.
 11. The communication device of claim 10, wherein the processor is configured to apply, after the construction of the 3D channel tensor, a transformation/compression with respect to all three dimensions of the 3D channel tensor

of dimension N_(r)×N_(t)×S represented by a (column-wise) Kronecker product as

=

_((1,2,3))(

) ${{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2,3})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,n_{r},n_{t},s} \otimes b_{2,n_{r},n_{t},s} \otimes b_{1,n_{r},n_{t},s}}}}}}}},$ where b_(1,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size N_(r)×1 with respect to the first dimension of the 3D channel tensor

, selected from a codebook matrix Ω₁; b_(2,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size N_(t)×1 with respect to the second dimension of the 3D channel tensor

, selected from a codebook matrix Ω₂; b_(3,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size S×1 with respect to the third dimension of the 3D channel tensor

, selected from a codebook matrix Ω₃; γ_(n) _(r) _(,n) _(t) _(,s) is the transformed/compressed channel coefficient associated with the vectors b_(1,n) _(r) _(,n) _(t) _(,s), b_(2,n) _(r) _(,n) _(t) _(,s), and b_(3,n) _(r) _(,n) _(t) _(,s), and N′_(r), N′_(t), and S′ (N′_(r)≤N_(r), N′_(t)≤N_(t), S′≤S) represents the value of the first, second and third dimension of the transformed/compressed 3D channel tensor

, respectively, or the space dimensions of the 3D channel tensor, represented by ${\mathcal{H} = {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,s} \otimes b_{2,n_{r},n_{t},s} \otimes b_{1,n_{r},n_{t},s}}}}}}}},$ where b_(3,s) is a transformation vector of all zeros with the s-th element being one, b_(2,n) _(r) _(,n) _(t) _(,s) is a vector of size N_(t)×1 with respect to the second dimension of the 3D channel tensor

, selected from a codebook matrix Ω₂, and b_(1,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size N_(r)×1 with respect to the first dimension of the 3D channel tensor

, selected from a codebook matrix Ω₁, N′_(r)≤N_(r), N′_(t)≤N_(t), S′=S, or the frequency dimension of the 3D channel tensor, represented by

=

₍₃₎(

), ${{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{(3)}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\gamma_{n_{r},n_{t},s}{b_{3,n_{r},n_{t},s} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}},$ where b_(2,n) _(r) is a transformation vector of all zeros with the n_(r)-th element being one, b_(1,n) _(t) is a vector of all zeros with the n_(t)-th element being one, and b_(3,n) _(r) _(,n) _(t) _(,s) is a transformation vector of size S×1 with respect to the third dimension of the 3D channel tensor

, selected from a codebook matrix Ω₃, N′_(r)=N_(r) and N′_(t)=N_(t), S′≤S.
 12. The communication device of any one of claim 11, wherein the 3D transformation function

_((1,2,3))(·) is given by a two-dimensional Discrete Cosine transformation (2D-DCT) and a 1D-DFT transformation with respect to the space and frequency dimension of the channel tensor, respectively, the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices, and the codebook matrix Ω₃ is given by an oversampled DFT matrix, or the 3D transformation function

_((1,2,3))(·) is given by a 3D-DFT transformation and the codebook matrices Ω_(n), n=1,2,3 are given by oversampled DFT matrices, or the 2D transformation function

_((1,2))(·) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices, or the 1D transformation function

₍₃₎(·) is given by a 1D-DFT transformation and the codebook matrix Ω₃ is given by an oversampled DFT matrix.
 13. The communication device of claim 10, wherein the processor is configured to apply, after the construction of the 4D channel tensor a transformation/compression with respect to all four dimensions of the channel tensor, represented by a (column-wise) Kronecker product as

=

_((1,2,3,4))(

), ${{{vec}(\mathcal{H})} = {{{vec}\left( {\mathcal{F}_{({1,2,3,4})}\left( \hat{\mathcal{H}} \right)} \right)} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r},n_{t},s,d} \otimes b_{1,n_{r},n_{t},s,d}}}}}}}}},$ where b_(1,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size N_(r)×1 with respect to the first dimension of the channel tensor

, selected from a codebook matrix Ω₁; b_(2,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size N_(t)×1 with respect to the second dimension of the channel tensor

, selected from a codebook matrix Ω₂; b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size S×1 with respect to the third dimension of the channel tensor

, selected from a codebook matrix Ω₃; b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size D×1 with respect to the fourth dimension of the channel tensor

, selected from a codebook matrix Ω₄; γ_(n) _(r) _(,n) _(t) _(,s,d) is the transformed/compressed channel coefficient associated with the vectors b_(1,n) _(r) _(,n) _(t) _(,s,d), b_(2,n) _(r) _(,n) _(t) _(,s,d), b_(3,n) _(r) _(,n) _(t) _(,s,d), and b_(4,n) _(r) _(,n) _(t) _(,s,d) and N′_(r), N′_(t), S′ and D′ (N′_(r)≤N_(r), N′_(t)≤N_(t), S′≤S, D′≤D) represents the value of the first, second, third and fourth dimension of the transformed/compressed channel tensor

, respectively, or the space dimensions of the 4D channel tensor, represented by ${\mathcal{H} = {\mathcal{F}_{({1,2})}\left( \hat{\mathcal{H}} \right)}},{{{vec}(\mathcal{H})} = {\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,d} \otimes b_{3,s} \otimes b_{2,n_{r},n_{t},s,d} \otimes b_{1,n_{r},n_{t},s,d}}}}}}}},$ where b_(4,d) is a transformation vector of all zeros with the d-th element being one, b_(3,s) is a transformation vector of all zeros with the s-th element being one, b_(2,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size N_(t)×1 with respect to the second dimension of the channel tensor

, selected from a codebook matrix Ω₂, and b_(1,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size N_(r)×1 with respect to the first dimension of the channel tensor

, selected from a codebook matrix Ω₁ and N′_(r)≤N_(r), N′_(t)≤N_(t), S′=S and D′=D, or the frequency and time dimensions of the 4D channel tensor, represented by

=

_((3,4))(

), ${{{vec}(\mathcal{H})} = {{{{vec}\left( {\mathcal{F}_{({3,4})}\left( \hat{\mathcal{H}} \right)} \right)} =}{\sum\limits_{n_{r} = 0}^{N_{r}^{\prime} - 1}{\sum\limits_{n_{t} = 0}^{N_{t}^{\prime} - 1}{\sum\limits_{s = 0}^{S^{\prime} - 1}{\sum\limits_{d = 0}^{D^{\prime} - 1}{\gamma_{n_{r},n_{t},s,d}{b_{4,n_{r},n_{t},s,d} \otimes b_{3,n_{r},n_{t},s,d} \otimes b_{2,n_{r}} \otimes b_{1,n_{t}}}}}}}}}},$ where b_(4,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size D×1 with respect to the fourth dimension of the channel tensor

,selected from a codebook matrix Ω₄, b_(3,n) _(r) _(,n) _(t) _(,s,d) is a transformation vector of size S×1 with respect to the third dimension of the channel tensor

, selected from a codebook matrix Ω₃, b_(2,n) _(r) is a transformation vector of all zeros with the n_(r)-th element being one, and b_(1,n) _(t) is a transformation vector of all zeros with the n_(t)-th element being one and N′_(r)=N_(r) N′_(t)=N_(t), S′≤S and D′≤D.
 14. The communication device of claim 13, wherein the 4D transformation function

_((1,2,3,4))(·) is given by a 4D-DFT transformation and the codebook matrices Ω_(n), n=1,2,3,4 are given by oversampled DFT matrices, or the 2D transformation/compression function

_((3,4))(·) is given by a 2D-DFT and the codebook matrices Ω_(n), n=3,4 are given by oversampled DFT matrices, or the 2D transformation function

_((1,2))(·) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices Ω_(n), n=1,2 are given by oversampled DCT matrices, or a 1D transformation function F₍₃₎(·) is given by a 1D-DFT transformation and the codebook matrix Ω₃ is given by an oversampled DFT matrix, or a 1D transformation function F₍₄₎(·) is given by a 1D-DFT transformation and the codebook matrix Ω₄ is given by an oversampled DFT matrix.
 15. The communication device of claim 11, wherein the processor is configured to select the transformation vectors from codebook matrices Ω_(n), and store the selected indices in a set

of g-tuples, where g refers to the number of transformed dimensions of the channel tensor, and report the set

as a part of the CSI report to the transmitter.
 16. The communication device of claim 8, wherein the oversampling factors of the codebooks are configured via the CSI report configuration, or via higher layer or physical layer by the transmitter, pre-determined and known at the communication device.
 17. The communication device of claim 1, wherein the communication device is configured with one or more scalar codebooks for the quantization of each entry of each basis vector of the plurality of dominant principal components of the channel tensor or the compressed channel tensor and the singular values or high order singular values, or with one or more unit-norm vector codebooks for the quantization of each basis vector of the plurality of dominant principal components of the channel tensor or the compressed channel tensor, and wherein the communication device selects for each basis vector a codebook vector to represent the basis vector, and the communication device is configured to report the indices corresponding to the selected entries in the scalar or vector codebook as a part of the CSI report to the transmitter.
 18. A transmitter in a wireless communication system, the transmitter comprising: an antenna array comprising a plurality of antennas for a wireless communication with one or more communication devices of claim 1 for providing a channel state information, CSI, feedback to the transmitter; and a precoder connected to the antenna array, the precoder to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams, a transceiver configured to transmit, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS, and downlink signals comprising the CSI-RS configuration; and receive uplink signals comprising a plurality of CSI reports comprising an explicit CSI from the communication device; and a processor configured to construct a precoder matrix applied on the antenna ports using the explicit CSI.
 19. A wireless communication network, comprising: at least one communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising: a transceiver configured to receive, from a transmitter, a radio signal via a radio time variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, and a processor configured to estimate the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, construct a frequency-domain channel tensor using the CSI estimate, perform a higher-order principal component analysis, HO-PCA, on the channel tensor, identify a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and report to the transmitter the explicit CSI comprising the dominant principal components of the channel tensor, and at least one transmitter comprising: a plurality of antennas for a wireless communication with the communication device for providing a channel state information, CSI, feedback to the transmitter; and a precoder connected to the antenna array, the precoder to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams, a transceiver configured to transmit, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS, and downlink signals comprising the CSI-RS configuration; and receive uplink signals comprising a plurality of CSI reports comprising an explicit CSI from the communication device; and a processor configured to construct a precoder matrix applied on antenna ports using the explicit CSI.
 20. The wireless communication network of claim 19, wherein the communication device and the transmitter comprises one or more of: a mobile terminal, or stationary terminal, or cellular IoT-UE, or an IoT device, or a ground based vehicle, or an aerial vehicle, or a drone, or a moving base station, or road side unit, or a building, or a macro cell base station, or a small cell base station, or a road side unit, or a UE, or a remote radio head, or an AMF, or an SMF, or a core network entity, or a network slice as in the NR or 5G core context, or any transmission/reception point (TRP) enabling an item or a device to communicate using the wireless communication network, the item or device being provided with network connectivity to communicate using the wireless communication network.
 21. A method for transmitting in a wireless communication system comprising a communication device of claim 1 and a transmitter, the method comprising: transmitting, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS configuration, and downlink signals comprising the CSI-RS configuration; receiving, at the transmitter, uplink signals comprising a plurality of CSI reports comprising an explicit CSI from the communication device; constructing a precoder matrix for a precoder connected to an antenna array comprising a plurality of antennas; applying the precoder matrix on antenna ports using the explicit CSI so as to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams.
 22. A method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising: receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, performing a high-order principal component analysis, HO-PCA, on the channel tensor, identifying a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and reporting the explicit CSI comprising the dominant principal components of the channel tensor from the communication device to the transmitter. 